Abstract
In this thesis we study two uncertainty principles from the perspective of timefrequency analysis. The first part, to which a significant portion is dedicated,
is concerned with deriving an analog in the joint time-frequency representation
of the pre-established fractal uncertainty principle for a function and its Fourier
transform. This study is motivated by, and the subsequent results substantiate,
the fundamental idea that such analogs should exist, where the uncertainty principles cannot be avoided by a change of representation. The second part is focused
on Hardy’s uncertainty principle in the joint representation, where we utilize this
statement to reproduce and derive new uniqueness results for the solution of the
Schrödinger equation. This showcases that uncertainty principles in the joint representation are not only an interesting object of study as analogous statements,
but, within their own right, might contain applications beyond their initial scope
of time-frequency analysis.
Sammendrag
I denne avhandlingen studerer vi to usikkerhetsrelasjoner fra perspektivet tidsfrekvensanalyse. Del 1, hvor en betydelig andel er dedikert, omhandler å utlede et
analogt resultat i felles tidsfrekvensrepresentasjon til det allerede etablerte fraktale usikkerhetsprinsippet for en funksjon og dens Fourier-transformasjon. Denne
studien er motivert av, og de påfølgende resultatene underbygger, den fundamentale ideen at slike analoger burde eksistere, hvor usikkerhetsrelasjonene ikke kan
omgås ved å bytte representasjon. Del 2 fokuserer på Hardys usikkerhetsprinsipp i
felles representasjon, hvor vi anvender prinsippet til å reprodusere samt utlede nye
unikhetsresultater for løsningen av Schrödinger-likningen. Dette viser at usikkerhetsrelasjoner i felles representasjon ikke bare er interessante å studere som analoge
setninger, men kan i seg selv inneholde anvendelser utover deres opprinnelig omfang i tidsfrekvensanalyse.
iii
Preface
This thesis is submitted in partial fulfillment of the requirements for the degree of
Philosophiae Doctor (PhD) in Mathematical Sciences at the Norwegian University
of Science and Technology (NTNU), Trondheim, Norway. The research presented
here has been primarily funded by The Research Council of Norway, and was
conducted at the Department of Mathematical Sciences in the period 2019–2023,
under supervision of Prof. Eugenia Malinnikova and Prof. Franz Luef.
The thesis is divided into two parts – an introductory part, Part I, and a
research part, Part II. In the introduction relevant background theory is provided
in a manner so to motivate the subsequent research. The introduction is concluded
with a brief summary of each of the four research papers that make up Part II.
All of the four papers have been published in peer-reviewed research journals. The
papers appear as in the published version, where any minor changes or additions
are always specified.
Acknowledgements
First and foremost, I would like to express my most sincere gratitude to my main
supervisor, Eugenia Malinnikova, who introduced me to the field of time-frequency
analysis as a master’s student and who has stuck with me all these years. I deeply
value all the mathematical discussions we’ve had, from which I have personally
benefited immensely and that have helped shape the ideas of the thesis. Eugenia
has always taken a great interest in my work, and her guidance has been nothing
short of excellent, even at the distance Trondheim–Stanford.
I would also like to thank my co-supervisor Franz Luef, who has been my main
mathematical support in Trondheim. Franz has always been a great source of
encouragement, and I appreciate all of our conversations and his insightful input.
During my PhD, I’ve had the chance to visit and work at Stanford University
on two separate occasions. The Department of Mathematics at Stanford has been
most hospitable, and I truly appreciate the opportunity in getting to know the
mathematical community overseas.
I am truly grateful for the many mathematical colleagues I’ve come to know
over the years, both in Trondheim, at Stanford and elsewhere. They have made
my work experience far more enjoyable, and have on many occasions turned into
close friends. A special shout-out to my past and present office mates at the tenth
floor.
Finally, thank you to all my friends and family who I wouldn’t want to be
without. You help me put everything into perspective.
Helge Knutsen
Trondheim, March 2023
v
Contents
Abstract
iii
Preface
v
I
1
Introduction
1 Preliminaries and motivation
1.1 Basics in time-frequency analysis . . . . . . . . . . . . . . .
1.1.1 Joint representations: Definitions and domains . . .
1.1.2 Time-frequency localization operators . . . . . . . .
1.1.3 Discretization: Gabor frames and Gabor multipliers
1.2 The fractal uncertainty principle . . . . . . . . . . . . . . .
1.2.1 Precise formulation in the separate representation .
1.2.2 Analogous results in the joint representation . . . .
1.3 The dynamical Hardy’s uncertainty principle . . . . . . . .
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2 Summary of papers
2.1 Paper A: Daubechies’ time-frequency localization operator on
Cantor type sets I . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Paper B: Daubechies’ time-frequency localization operator on
Cantor type sets II . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Paper C: A fractal uncertainty principle for the short-time Fourier
transform and Gabor multipliers . . . . . . . . . . . . . . . . . . .
2.4 Paper D: Notes on Hardy’s uncertainty principle for the Wigner
distribution and Schrödinger evolutions . . . . . . . . . . . . . . .
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II Research papers
23
A Daubechies’ time-frequency localization operator on Cantor type
sets I
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.1 Fourier and short–time Fourier transform . . . . . . . . . .
A.2.2 Daubechies’ localization operator . . . . . . . . . . . . . . .
A.2.3 Spherically symmetric weight . . . . . . . . . . . . . . . . .
A.2.4 Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Examples of localization on spherically symmetric sets . . . . . . .
A.3.1 Localization on a ring: Asymptotic estimate . . . . . . . . .
A.3.2 Localization on set of infinite measure . . . . . . . . . . . .
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Contents
A.4 Localization on spherically symmetric Cantor set
A.4.1 Results: Bounds for the operator norm . .
A.4.2 Main strategy: Relative areas . . . . . . .
A.4.3 Proof of Theorem A.4.1 . . . . . . . . . .
A.4.4 Proof of Theorem A.4.2 . . . . . . . . . .
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Appendices
A.A Omitted proofs in Section A.2.3 . . . . . . . . . . . . . . . . . . . .
45
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B Daubechies’ time-frequency localization operator on Cantor type
sets II
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2.1 Daubechies’ localization operator . . . . . . . . . . . . . . .
B.2.2 Generalized Cantor set construction . . . . . . . . . . . . .
B.3 Localization on generalized spherically symmetric Cantor set . . .
B.3.1 Results: Bounds for the operator norm . . . . . . . . . . . .
B.3.2 Main tool: Relative areas . . . . . . . . . . . . . . . . . . .
B.3.3 Proof of Theorem B.3.1 . . . . . . . . . . . . . . . . . . . .
B.3.4 Proof of Theorem B.3.2 . . . . . . . . . . . . . . . . . . . .
B.3.5 Proof of Theorem B.3.3 (c):
Counterexample to precise asymptotic estimate . . . . . . .
B.4 Further generalizations: Indexed Cantor set . . . . . . . . . . . . .
B.4.1 Results: Sufficient decay conditions . . . . . . . . . . . . . .
B.4.2 Set-up and simple example . . . . . . . . . . . . . . . . . .
B.4.3 Proofs of Theorem B.4.1 and Theorem B.4.2 . . . . . . . .
Appendices
B.A Omitted proof in Section B.2.2: Weak subadditivy . . . . . . . . .
C A fractal uncertainty principle for the short-time Fourier
transform and Gabor multipliers
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2.1 The short-time Fourier transform and modulation spaces
C.2.2 From modulation spaces to Fock spaces . . . . . . . . . .
C.2.3 Gabor frames and Gabor multipliers . . . . . . . . . . . .
C.2.4 Porous sets and Cantor sets . . . . . . . . . . . . . . . . .
C.3 Fractal uncertainty principle in joint representation . . . . . . . .
C.3.1 Fractal uncertainty principle for Fock spaces . . . . . . .
C.3.2 Fractal uncertainty principle for modulation spaces . . . .
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Contents
C.4 Density of Cantor sets . . . . . . . . . . . . . . . . . . . . . . . . .
C.5 Fractal uncertainty principle for Gabor multipliers . . . . . . . . .
Appendices
C.A Complex interpolation in Fock space . . . . . . . . . . .
C.B Omitted proof: Simple porosity estimate of Cantor sets
C.C Subaveraging in Fock space . . . . . . . . . . . . . . . .
C.D Generalizations of the fractal uncertainty principle . . .
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D Notes on Hardy’s uncertainty principle for the Wigner
distribution and Schrödinger evolutions
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.2.1 Weyl quantization and the Schrödinger equation . . . . .
D.2.2 Symplectic mechanics . . . . . . . . . . . . . . . . . . . .
D.2.3 Hardy’s uncertainty principle and the Wigner distribution
D.3 Hardy’s uncertainty principle for quadratic Hamiltonians . . . . .
D.4 Examples of Schrödinger evolutions . . . . . . . . . . . . . . . . .
D.4.1 Free particle, harmonic oscillator and magnetic potential .
D.4.2 Systems based on positive definite matrices . . . . . . . .
Appendices
D.A Williamson’s diagonalization theorem
Bibliography
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ix
Part I
Introduction
Chapter 1
Preliminaries and motivation
As the title of the thesis suggests, the research presented is all, in some form,
centered on the uncertainty principles in signal analysis and more specifically timefrequency analysis. Informally, the uncertainty principles can be summarized as
the meta-theorem:
A signal cannot be arbitrarily well localized in both time and frequency.
There are many versions that all embody this idea, where perhaps the most widely
recognized version is Heisenberg’s uncertainty principle, which utilizes the familiar
notion of standard deviation. Together, these statements represent a fundamental
barrier with regard to optimal time-frequency localization. Questions related to
time-frequency localization will therefore often yield non-trivial and interesting
answers. In this thesis, these type of questions will be studied from the perspective
of joint time-frequency representations.
The remainder of the chapter is divided into three parts: We first provide a
brief introduction to the central concepts of time-frequency analysis in Section 1.1.
In Section 1.2 we present the original fractal uncertainty principle and motivate
our search for an analogous statement. Finally, in Section 1.3 we discuss Hardy’s
uncertainty principle and dynamical versions of this principle.
1.1
Basics in time-frequency analysis
In signal analysis we consider functions of the form f : R (or Rd ) → C, where we
oftentimes interpret f as a time-dependent signal, and we wish to analyze both the
time and frequency behaviour of said signal and their co-dependence. Provided f
is sufficiently nice, it is well known that the Fourier transform yields a frequency
representation of the signal, say fˆ, meaning we would initially study the pair
(f, fˆ). We refer to this pair as the separate time-frequency representation of the
signal, where the time and frequency content are described by separate functions,
connected by a transform. Although a very popular and classical approach, this
perspective does have some drawbacks with regards to the physical interpretation
3
Chapter 1. Preliminaries and motivation
of the time and frequency content. Namely, from the Fourier transform we can say
which frequencies feature but not when they feature. A more natural approach
would therefore be to transform the signal on R (or Rd ) into a function on R×R (or
Rd × Rd ), so that the time and frequency behaviour are described simultaneously.
Such representations are aptly referred to as joint or simultaneous time-frequency
representations, and it is the study of these representations which is the primary
focus of time-frequency analysis.
1.1.1
Joint representations: Definitions and domains
In this thesis we consider two such joint time-frequency representations, namely the
short-time Fourier transform (STFT) and the (cross-)Wigner distribution. Each
of these transforms are closely related to the regular Fourier transform, but takes
two functions f, g : Rd → C as input instead of one, so that evaluated at a point
(x, ω) ∈ Rd × Rd , the STFT reads
Z
Vg f (x, ω) :=
f (t)g(t − x)e−2πiω·t dt,
Rd
and in turn the cross-Wigner distribution reads
Z
W (f, g)(x, ω) :=
f (t + x/2)g(t − x/2)e−2πiω·t dt.
Rd
The above notation suggests that the signal f is the primary input in the STFT,
and we tend to think of g as a fixed window function. Conversely, for the crossWigner distribution, we oftentimes let f and g coincide, in which case, we refer
to W (f, f ) =: W f simply as the Wigner distribution of f . Nonetheless, it is easy
to verify that these are equivalent representations, meaning one can always be
expressed in terms of the other. Where they differ is the context in which they
are typically preferred and more widely used. E.g., in quantum mechanics, the
(cross-) Wigner distribution is often the preferred choice, while with regard to
time-frequency localization and localization operators, the STFT seems to be a
more natural candidate. This ”distinction” is in keeping with the choice of joint
representation in each of the papers that make up the thesis.
Proceeding, we focus on the STFT and its time-frequency interpretation. In
time-frequency analysis we consider two basic operators
Tx g(t) := g(t − x) and Mω g(t) := e2πω·t g(t),
that is, a time-shift and frequency modulation, which combined yield a single
time-frequency shift
π(x, ω)g(t) := Mω Tx g(t) = e2πω·t g(t − x).
Thus, the STFT Vg f (x, ω) can be more compactly expressed as a time-frequency
shift of the window, π(x, ω)g, acting as a linear functional on the signal f , i.e.,
Vg f (x, ω) = ⟨f, π(x, ω)g⟩.
4
(1.1.1)
1.1. Basics in time-frequency analysis
We note that for g ≡ 1, there is no time dependence in Vg f , and the STFT is
reduced to the regular Fourier transform. However, the motivation for including
a window function g is for π(x, ω)g to measure the contribution of the signal f
near time x and frequency ω. We would therefore typically consider some function g so that both g and ĝ are close to 1 near the origin and decay rapidly to
zero outside some small neighbourhood. The more concentrated our window is
near the origin, the sharper our time measurements will become – and similarly,
a more concentrated Fourier transform, yields sharper frequency measurements.
This immediately begs the question whether we could have instantaneous time and
frequency measurements. Unfortunately, such aspirations are meaningless once we
consult the uncertainty principles.
Similarly to regular Fourier analysis, we need to address for which signals (and
windows) the STFT is actually well-defined. A popular choice for both signal and
window space is L2 (Rd ), which, by Cauchy–Schwarz’ inequality, clearly makes Vg f
in (1.1.1) well-defined. Furthermore, for this function space the STFT satisfies the
following orthogonality relations
⟨Vg1 f1 , Vg2 f2 ⟩L2 (R2d ) = ⟨f1 , f2 ⟩⟨g1 , g2 ⟩ ∀ f1 , f2 , g1 , g2 ∈ L2 (Rd ).
Thus, whenever the window g is an L2 -function, the STFT is a continuous map
Vg : L2 (Rd ) → L2 (R2d ). In particular, if ∥g∥2 = 1, then Vg is an isometry, and we
have the reconstruction formula
Z
f=
Vg f (x, ω)π(x, ω)g dxdω,
(1.1.2)
R2d
where the integral is interpreted in the weak-sense. The reconstruction can also
be viewed as the composition Vg∗ ◦ Vg of the STFT Vg with its adjoint
Z
∗
2
2d
2
d
Vg : L (R ) → L (R ), F 7→
F (x, ω)π(x, ω)g dxdω.
R2d
From this perspective, we would refer to Vg as an analysis mapping – analyzing the
time-frequency content of a signal, and we would refer to Vg∗ as a synthesis mapping
Vg∗ – synthesizing the signal from its previous analysis. Although in general, the
analysis and synthesis window do not need to coincide, this will always be presumed
in our subsequent discussions on reconstruction and localization operators.
Another popular signal space are the so-called modulation spaces M p (Rd ), first
introduced and developed by Feichtinger for p = 1 in [34] and later for general
p ∈ [1, ∞] in [35]. Here we fix some window g ̸≡ 0 in the Schwartz’ class S (Rd ),
and we consider all signals f in the dual of tempered distributions S ′ (Rd ) such
that Vg f is an Lp -function, i.e.,
M p (Rd ) := f ∈ S ′ (Rd ) ∥Vg f ∥Lp (R2d ) < ∞ .
Note that these spaces are invariant for any particular choice of window function. In fact, any g ∈ S (Rd ) \ {0} induces equivalent norms on M p (Rd ), namely
∥f ∥M p := ∥Vg f ∥Lp (R2d ) , and turns the modulation space into a Banach space.
Notably for p = 2, we have the identification M 2 (Rd ) = L2 (Rd ).
In the subsequent Paper A and B, we restrict our attention to L2 -signals, and
later in Paper C, we also consider the more general family of M p -spaces.
5
Chapter 1. Preliminaries and motivation
1.1.2
Time-frequency localization operators
Oftentimes, we wish to analyze signals on specific time-frequency domains. In order
to do so, we consider a certain kind of operator, more commonly known as a timefrequency localization operator, whose purpose is to concentrate signals on these
domains. Depending on our perspective, be it the separate or joint representation,
different operators have been proposed.
In the separate representation, a popular choice is the composition operator
χW F χT : L2 (Rd ) → L2 (Rd ),
projecting first onto our time domain T and then onto our frequency domain W .
Here F denotes the regular Fourier transform, and χE denotes the characteristic
function of a set E ⊆ Rd . Historically, the study of this localization operator dates
back to the 1960’s, where Slepian, Landau and Pollak, in their celebrated series of
Bell Lab papers [79], [67], [68], considered time and frequency bands of the form
|T | |T |
|W | |W |
,
,
and W = −
.
T = −
2 2
2
2
This composition operator remains relevant and has been extensively studied, even
in the last couple of years. Notably the original fractal uncertainty principle is
formulated in terms of this particular operator, where T and W are now equipped
with a fractal structure (see Section 1.2).
In the joint representation, for L2 -signals, there is a natural way to define timefrequency localization operators based on reconstruction formula (1.1.2). This
approach was suggested by Daubechies in her 1988-paper [22], where, specifically,
we modify the STFT Vg f by a multiplicative weight function S ∈ L∞ (R2d ) before
reconstructing a time-dependent signal, i.e.,
Z
PSg f :=
S(x, ω) · Vg f (x, ω)π(x, ω)g dxdω
(1.1.3)
R2d
⇐⇒ ⟨PSg f, h⟩ = ⟨S · Vg f, Vg h⟩L2 (R2d ) ∀ h ∈ L2 (Rd ).
The purpose of the weight or symbol function S is to enhance features of the timefrequency plane that we wish to analyze. Typically, we consider symbols of the
form S = χΩ for some Ω ⊆ R2d , in which case we write PΩg := PχgΩ . For Ω = T ×W ,
we find that Daubechies’ operator more closely resembles an analogous operator
to the composition operator from the separate representation. – But unlike in the
separate representation, for some fixed time-frequency domain Ω, we obtain an
entire family of operators PΩg : L2 (Rd ) → L2 (Rd ) indexed by our choice of window
g ∈ L2 (Rd ).
In our analysis of Daubechies’ localization operator in Paper A, B and C we
shall almost exclusively focus on the normalized Gaussian window
2
ϕ0 (x) := 2d/4 e−πx , where x2 = x21 + · · · + x2d for x = (x1 , . . . , xd ).
(1.1.4)
Although this is a rather specific choice, the Gaussian is in some sense regarded as
the canonical window as it minimizes several of the uncertainty principles. From a
6
1.1. Basics in time-frequency analysis
practical point of view, the Gaussian also provides a useful structure to the STFT
and by extension the localization operator. This is described in more detail at the
end of Section 1.2 and in each of the ”Preliminaries” sections in the aforementioned
papers.
1.1.3
Discretization: Gabor frames and Gabor multipliers
So far we have considered reconstruction and localization based on continuous
samples over R2d of the time-frequency content of a signal f ∈ L2 (Rd ). While this
certainly is relevant within the theoretical framework, it is not always convenient
for practical purposes and real-world applications. Instead we would like to replace
2d
the continuous samples of Vg f (x, ω)
and π(x, ω)g over2d R by discrete samples
over some countable subset Λ = (xj , ωj ) j∈N ⊆ R , and construct discrete
alternatives to our previous formulas. The study of such discretizations is referred
to as Gabor analysis, which rests within the broader theory of frames.
In general, for a (separable) Hilbert space H, a frame is a countable family of
vectors {ϕk }k ⊆ H to which there exist two constants 0 < A ≤ B < ∞, i.e., frame
bounds, such that
X
A∥f ∥2H ≤
|⟨f, ϕk ⟩H |2 ≤ B∥f ∥2H ∀ f ∈ H.
k
If A = B, we say that the frame is tight, and if A = B = 1, we call it a Parseval
frame. In fact, any tight frame
{ϕk }k , may be turned into a Parseval frame by the
√
normalization, ϕk 7→ ϕk / A. To the frame {ϕk }k , we associate an analysis and
synthesis mapping, namely
X
T : H → ℓ2 , f 7→ (⟨f, ϕk ⟩H )k and T ∗ : ℓ2 → H, (ck )k 7→
ck ϕk ,
k
which combined yields the so-called frame operator
X
S = T ∗ T : H → H, f 7→
⟨f, ϕk ⟩H ϕk .
k
From its composition and the frame condition, we easily deduce that S is bounded
with ∥S ∥H ≤ B (with equality if B is optimal), positive and self-adjoint. Furthermore, the frame operator is invertible, which, in turn, yields the identities
X
X
f=
⟨f, ϕk ⟩H S −1 ϕk =
⟨f, S −1 ϕk ⟩H ϕk .
k
k
In thePcase of the Parseval frame, the frame operator S equals the identity, i.e.,
f =
k ⟨f, ϕk ⟩H ϕk . Thus, in terms of reconstruction, a frame displays similar
behaviour to an orthonormal basis. However, the vectors {ϕk }k are not required to
be orthogonal, not even linearly independent. Instead, frames oftentimes represent
overcomplete systems or systems with a certain amount of redundancy, meaning
we can remove a certain subset {ϕkj }N
j=1 (where N is finite or infinite) from {ϕk }k
7
Chapter 1. Preliminaries and motivation
and still maintain a frame. With this terminology, an orthonormal basis represents
a (Parseval) frame with no redundancy.
In Gabor analysis, we consider
systems of the form {π(λ)g}λ∈Λ , where g is
some fixed window and Λ = (xj , ωj ) j ⊆ R2d is a lattice of sampling points, e.g.,
a rectangular lattice aZd × bZd for a, b > 0. To each such lattice we associate a
connected neighbourhood of the origin called the fundamental domain DΛ such
that
∪λ∈Λ (λ + DΛ ) = R2d and |(λ + DΛ ∩ (ξ + DΛ )| = 0 whenever λ ̸= ξ.
The shape of DΛ describes the structure of Λ, and the size |DΛ | describes the
density of the lattice, i.e., the density of sampling points. We now ask whether
this collection of time-frequency shifts {π(λ)g}λ∈Λ can in fact constitute a frame.
Infamously, for the Gaussian window ϕ0 in dimension d = 1, there is a simple
density criterion on the lattice, discovered independently by Lyubarskii [72] and
Seip and Wallsten [76], [77]. Their result states that {π(λ)ϕ0 }λ∈Λ is a frame if and
only if |DΛ | < 1. For other windows, however, Janssen shows in [60] that precise
lattice conditions can be far more complicated.
p
Interestingly enough, with the normalization |DΛ |{π(λ)g}λ∈Λ , we recognize
that the frame operator
X
SΛg : L2 (Rd ) → L2 (Rd ), f 7→ |DΛ |
Vg f (λ)π(λ)g
(1.1.5)
λ∈Λ
is a Riemann sum of the reconstruction
formula (1.1.2). Thus, presuming ∥g∥2 = 1
p
and considering a sequence |DΛn |{π(λ)g}λ∈Λn where |DΛn | → 0 as n → ∞, we
could expect the frame operator SΛg n to converge weakly to the identity. In fact,
Weisz has proven in [84] that this intuition is correct, at least for square lattices
Λn = n1 (Zd × Zd ) and windows g ∈ M 1 (Rd ).
Just as the frame operator is a discrete approximation to the reconstruction
formula (1.1.2) (with equality in the Parseval frame case), the Gabor multiplier
represents a discrete alternative to Daubechies’ localization operator (1.1.3). The
Gabor multiplier, which is a useful tool in real-world applications such as object
detection and signal compression (see [26], [81]), is constructed by modifying the
summation in (1.1.5) by a bounded symbol b : Λ → C, so that
X
g
GΛ,b
: L2 (Rd ) → L2 (Rd ), f 7→ |DΛ |
b(λ) · Vg f (λ)π(λ)g.
(1.1.6)
λ∈Λ
2d
For localization on a specific subset Ω ⊆ R , there are two natural options for the
symbol b to mimic the characteristic function χΩ . One possibility is to let b(λ)
measure the portion of λ + DΛ containing Ω, i.e.,
b1 (λ) =
|Ω ∩ (λ + DΛ )|
∈ [0, 1].
|DΛ |
Alternatively, we could simply consider whether there is a non-zero overlap, i.e.,
(
1, |Ω ∩ (λ + DΛ )| > 0,
b2 (λ) =
0, otherwise
8
1.2. The fractal uncertainty principle
Evidently, the latter option simply amounts to restricting the summation in (1.1.5)
g
to a subset ΛΩ ⊆ Λ, which, unsurprisingly, makes GΛ,b
easier to work with than
2
g
g
g
GΛ,b1 . Since we also have that ∥GΛ,b1 ∥op ≤ ∥GΛ,b2 ∥op , we shall focus on the latter
option in Paper C when we finally discuss the fractal uncertainty principle for
Gabor multipliers.
1.2
The fractal uncertainty principle
The fractal uncertainty principle (FUP) was first introduced by Dyatlov and Zahl
in their 2016-paper [29] and further developed in the works [12], [28] and [61]
by Dyatlov, Bourgain, Jin and Zhang. Here the FUP appears in the context of
quantum chaos and mircolocal analysis, where it has been successfully applied to
problems related to spectral gaps on non-compact hyperbolic surfaces and lower
bounds on mass of eigenfunctions. For a review on these applications, we reference
Dyatlov’s detailed introduction to the FUP [27] Section III. In this thesis, we
approach this uncertainty principle as a non-trivial localization result in signal
analysis. As the name suggests, the FUP is concerned with localization on timefrequency domains that have a fractal structure. Informally speaking, it states
that
a signal cannot be concentrated near a fractal set in both time and
frequency.
Since the original statement is only formulated in terms of a signal and its Fourier
transform, this inspires us to search for similar results in the joint time-frequency
representation. Our search is motivated by the underlying philosophy, or the metatheorem, that the uncertainty principles are fundamental in the sense that
the uncertainty principles cannot be avoided by a change in time-frequency
representation.
The idea that analogs of uncertainty principles should exist is well-documented by
several examples. We list two such examples to illustrate this point:
Example 1.2.1. (Finite support in time and frequency)
In the 1985-paper [10], Benedicks showed that a signal and its Fourier transform
cannot both be supported on a set of finite measure.
Theorem 1.2.1. Suppose f ∈ L2 (Rd ) such that |suppf |·|suppfˆ| < ∞, then f ≡ 0.
It was then conjectured by Folland and Sitaram in their 1997-survey paper
[40] that a similar result should hold for the Wigner distribution (or equivalently
the STFT). The following year, a version of Benedicks’ theorem was proven in
the joint representation, credited Jaming, Janssen and Wilczok in [58], [59], [85],
respectively.
Theorem 1.2.2. Suppose f, g ∈ L2 (Rd ) such that |suppVg f | < ∞, then Vg f ≡ 0
so f ≡ 0 or g ≡ 0.
9
Chapter 1. Preliminaries and motivation
Example 1.2.2. (Lower bound measure of time-frequency domains)
In their 1989-paper [25], Donoho and Stark presented a comparatively more
quantitative result than Benedicks’ theorem, concerning lower bound measures of
time and frequency sets at a certain level of signal-concentration.
Theorem 1.2.3. Suppose ∥f ∥2 = 1, and suppose T, W ⊆ Rd are two measurable
subsets such that ∥f −χT f ∥2 < ϵT and ∥fˆ−χW fˆ∥2 < ϵW for some 0 < ϵT , ϵW ≤ 12 .
Then
|T | · |W | ≥ (1 − ϵT − ϵW )2 .
In the joint representation, Lieb proved in his 1990-paper [69] a general integral
inequality, which in terms of the STFT reads
∥Vg f ∥Lp (R2d )
d/p
2
∥f ∥2 ∥g∥2 ∀ f, g ∈ L2 (Rd ) and p ≥ 2.
≤
p
From this inequality, an analog to Donoho-Stark’s uncertainty principle follows,
commonly referred to as Lieb’s uncertainty principle (see Theorem 3.3.3 in [47]).
Theorem 1.2.4. Suppose f, g ∈ L2 (Rd ) \ {0}, and suppose that Ω ⊆ R2d is
a measurable subset such that ∥χΩ · Vg f ∥2L2 (R2d ) ≥ (1 − ϵ)∥Vg f ∥2L2 (R2d ) for some
0 < ϵ < 1. Then
p
|Ω| ≥ sup(1 − ϵ) p−2
p>2
2d
p p−2
2
.
While the above Lieb’s uncertainty principle is not claimed to be sharp, Nicola
and Tilli recently proved in [73] (see Theorem 1.2 for d = 1 and Corollary 4.2 for
d > 1) a sharp version
for the Gaussian window g = ϕ0 . In the case d = 1, we
have that |Ω| ≥ ln 1ϵ with equality if and only if the signal f = cπ(x0 , ω0 )ϕ0 for
some c ∈ C \ {0}, and Ω is a ball centered at (x0 , ω0 ) ∈ R2 .
What is interesting to note about the two analogs described above, is how
they differ in their derivation. The proof of the analog of Benedicks’ theorem
for the STFT utilizes the original statement for the separate representation. In
contrast, the proof of Lieb’s uncertainty principle does not rely on Donoho-Stark’s
uncertainty principle. As we shall soon discover in Paper A, B and C, our versions
of the FUP in the joint context belong to the second category of analogs. The
original formulation of the FUP in the separate representation serves as inspiration,
but is not directly involved in our proofs.
1.2.1
Precise formulation in the separate representation
In order to formulate the FUP precisely, we first need to clarify what we mean
by ”fractal” or ”fractal structure”. Following Dyatlov’s survey on the FUP in
[27], fractal structures are described broadly, either in terms of δ-regularity or νporosity for some 0 < δ < 1 and 0 < ν < 1 (see Def. 2.2 and Def. 2.7). Since these
10
1.2. The fractal uncertainty principle
are equivalent concepts in the context of the FUP, we shall focus on ν-porosity
as it is the less abstract of the two and is easier to work with. While originally
formulated for subsets of R, we have adjusted the definition to higher dimensions.
Def. 1.2.1. (ν-porosity) Suppose X ⊆ Rd is non-empty closed. Suppose 0 < ν < 1
and 0 ≤ αmin ≤ αmax ≤ ∞. We say that X is ν-porous on scales αmin to αmax if
for every ball Br (x) of radius r ∈ [αmin , αmax ] there exists a ball Bνr (y) ⊆ Br (x)
of radius νr such that |X ∩ Bνr (y)| = 0.
From the above definition, we read that a porous set X ⊆ Rd maintains certain
gaps or pores within its scale bounds. This is similar to and generalizes the notion
of a set with repeating patterns under scaling. The ”fractal set” is understood as
a family of porous sets X = X(h) that all share the same porosity constant ν > 0
but where the lower scale bound varies between 0 < h ≤ 1. For such a family
of sets, we think of the fractal structure being realized when h → 0. The FUP
is then formulated for such h-dependent families T (h), W (h) of porous sets in R
(under some additional conditions), and gives a norm estimate of the composition
operator χW (h) F χT (h) .
Theorem 1.2.5. (FUP in separate representation; Theorem 2.16 in [27]) Let
0 < h ≤ 1 be a continuous parameter, and suppose T (h), W (h) ⊆ [0, h−1 ] are hdependent families that are ν-porous on scales h to h−1 . Then there exist constants
C, β > 0 only dependent on ν such that
∥χW (h) F χT (h) ∥L2 (R)→L2 (R) ≤ Chβ ∀ 0 < h ≤ 1.
Remark. In [27], there are no explicit estimates for the exponent β > 0, beyond
the trivial lower bound β > max{0, β0 } obtained from volume estimates of |T (h)|
and |W (h)|, where
p
∥χW (h) F χT (h) ∥L2 (R)→L2 (R) ≤ C |T (h)| · |W (h)| ≤ Chβ0 .
For improved, explicit estimates of β > 0, we refer to the recent paper [61].
Since the subsets T (h), W (h) in the above statement are possibly unbounded
as h → 0, this begs the question whether the associated measures can also be
unbounded. In fact, depending on our choice of ν > 0, there are ν-porous families
T (h), W (h) ⊆ [0, h−1 ] such that |T (h)|, |W (h)| → ∞ as h → 0. We illustrate this
non-trivial aspect of the FUP with a brief discussion of the Cantor set constructions. These constructions represent a popular and easy to understand class of
porous sets based on discrete iterations n = 0, 1, 2, . . .
Example 1.2.3. (Cantor sets) Fix some integer M > 1, called the base, and a
non-empty subset A ⊊ {0, 1, . . . , M − 1}, known as an alphabet. The n-iterate (or
n-order) discrete Cantor set with base M and alphabet A refers to a subset of
{0, 1, . . . , M n − 1} given by


n−1

X
Cn(d) (M, A ) =
aj M j aj ∈ A for j = 0, 1, . . . , n − 1 .


j=0
11
Chapter 1. Preliminaries and motivation
We acknowledge that these discrete constructions merit their own discussion, where
there actually is an FUP solely based on discrete Cantor sets, now in terms of the
discrete Fourier transform. We refer the reader to Section IV in [27] and the
recent developments in [16]. For our purpose, however, we utilize the discrete
Cantor iterates to define its ”continuous” counterpart based in the interval [0, L].
Namely, the n-iterate Cantor set with base M and alphabet A is given by
[
[xLM −n , (x + 1)LM −n ]
Cn (L, M, A ) :=
(d)
x∈Cn (M,A )

=
[
L
aj ∈A
n
X
j=1
aj M
−j
,L
n
X
(1.2.1)

aj M
−j
+ LM
−n 
⊆ [0, L].
j=1
Notably, for M = 3 and A = {0, 2}, we recognize the infamous mid-third Cantor
set construction. Presuming L > 0 is fixed throughout the iterations, we obtain a
nested sequence Cn ⊇ Cn+1 , and the (limit) Cantor set C (L, M, A ) is defined as
the intersection of all iterates. If we let |A | denote the cardinality of the alphabet,
the measure of each iterate reads
n
|A |
L for n = 0, 1, 2, . . . ,
|Cn (L, M, A )| =
M
meaning the Cantor set itself has measure zero.
As previously mentioned, the Cantor set constructions are examples of ν-porous
sets. In particular, we have a simple porosity estimate (see Appendix C.B), which
states that
Cn (L, M, A ) is ν-porous on scales LM −n+1 to ∞ for any ν ≤ 12 M −2 .
Thus, we may restate Theorem 1.2.5 for the family of Cantor constructions, where
the continuous lower scale bound 0 < h ≤ 1 is now replaced by the discrete
iterations n = 0, 1, 2, . . . such that h ∼ LM −n . Since we presume the porous sets
belong to the interval [0, h−1 ], we can also choose the initial interval length L > 0
accordingly, i.e., L ∼ h−1 . From here, we obtain the interval condition
n
L∼M2.
(1.2.2)
In this case we no longer have a nested sequence
of Cantor iterates. Rather, we
have an unbounded sequence of iterates Cn (L(n), M, A ) n whose measure is of
the form
n
|A |
|Cn (L(n), M, A )| ∼ √
for n = 0, 1, 2, . . .
M
√
This means, whenever |A | > M (such as for the mid-third Cantor set), the
measure |Cn (L(n), M, A )| will diverge to infinity as n → ∞ all the while
∥χCn (... ) F χCn (... ) ∥L2 (R)→L2 (R) → 0.
12
1.2. The fractal uncertainty principle
1.2.2
Analogous results in the joint representation
Similar to the FUP in the separate representation, our analogs for the STFT come
in the form of a norm estimate but where the composition operator χW F χT is
replaced by Daubechies’ localization operator PΩg . Specifically, we aim to estimate
the operator norm
Z
∥PΩg ∥op = sup ∥PΩg ∥2 = sup
|Vg f (x, ω)|2 dxdω,
∥f ∥2 =1
∥f ∥2 =1
Ω
where Ω ⊆ R2d is a fractal type set. Although there is an additional degree
of freedom in the choice of window g, our analysis is focused on the normalized
Gaussian window g = ϕ0 . As previously remarked, the Gaussian is the (unique)
minimizer of several uncertainty principles, and can therefore be regarded as a
canonical window choice. In fact, the original paper [22] by Daubechies is focused
on the Gaussian window. Moreover, she considers symbol functions with a radial
symmetry, which in our case translates to localization on subsets of the form
Ωρ1 ×···×ρd := Ωρ1 × · · · × Ωρd ⊆ R2d , where for each j = 1, 2, . . . , d
Ωρj = (xj , ωj ) ∈ R2 x2j + ωj2 ∈ ρj ⊆ R2 for some ρj ⊆ R+ .
(1.2.3)
The radial assumption is effective as it yields a known eigenbasis for PΩϕρ0 ×···×ρ ,
1
d
as well as a complete description of the associated eigenvalues.
Theorem 1.2.6. (Daubechies [22]) Suppose Ωρ1 ×···×ρd ⊆ R2d is of the form
(1.2.3), and suppose the window equals the normalized Gaussian ϕ0 in (1.1.4).
Then the eigenfunctions of Daubechies’ operator PΩϕρ0 ×···×ρ : L2 (Rd ) → L2 (Rd )
1
d
are the d-dimensional Hermite functions
H(k) (x) =
d
Y
Hkj (xj ) with k = (k1 , . . . , kd ) ∈ Nd0 and
j=1
21/4
Hkj (t) = p
kj !
kj
k
2 d j
2
1
− √
eπt kj (e−2πt ).
dt
2 π
The associated eigenvalues PΩϕρ0
1 ×···×ρd
λ(k) =
d
Y
j=1
H(k) = λ(k) H(k) are given by
Z
λkj with λkj =
π·ρj
rkj −r
e dr.
kj !
(1.2.4)
Since the operator norm of any self-adjoint compact operator coincides with the
largest eigenvalue (in absolute value), it suffices, in the radial case, to estimate the
supremum of the eigenvalue-formulas (1.2.4). With this powerful tool available, we
have in Paper A and B considered localization on radially symmetric sets Ωρ ⊆ R2
for unidimensional signals in L2 (R). – Note that these results are easily extended
to higher dimensions d > 1 by the factorization in Theorem 1.2.6. In particular,
we consider radial Cantor iterates based in the disk of radius R > 0 given by
Cn (R, M, A ) := (x, ω) ∈ R2 x2 + ω 2 ∈ Cn (R2 , M, A ) ⊆ R2 .
13
Chapter 1. Preliminaries and motivation
The associated eigenvalues of PCϕn0 (R,M,A ) then read
λk = λk (Cn (R, M, A )) =
Z
Cn (πR2 ,M,A )
rk −r
e dr for k = 0, 1, 2, . . .
k!
As a first analog of the FUP, we derive an upper bound asymptote for the operator
norm valid for all πR2 ≤ M n . We summarize the results under the condition
n
πR2 ∼ M 2 , mimicking condition (1.2.2) for the regular Cantor iterates.
Theorem 1.2.7. (FUP for radial Cantor sets) Suppose that the radius R depends
n
on the iterate n such that πR2 ∼ M 2 . Then there exists a finite constant C > 0
only dependent on M and |A | such that
∥PCϕn0 (R(n),M,A ) ∥op
≤C
|A |
M
n2
for n = 0, 1, 2, . . .
Furthermore, the above asymptote is precise for
• the mid-third Cantor iterates, where M = 3, A = {0, 2}, and
• the canonical alphabet A = {0, 1, . . . , |A |−1} at any alphabet size |A | < M .
Conversely, at any alphabet size |A | < M , there exist alphabets A such that the
asymptote is not precise.
Remark. In comparison with the original FUP in Theorem 1.2.5, our radial result
does not only contain an explicit estimate for the exponent β > 0 but also an
overall sharp estimate. However, as stated in the above theorem, the sharpness of
the exponent does not extend to every radial Cantor set construction.
For other, non-radial sets, we do not necessarily have explicit knowledge of
neither the eigenvalues nor the eigenfunctions. In this case, we consider the general
structure of the STFT Vϕ0 provided by the Gaussian window ϕ0 . Namely, for
signals in M p (Rd ) with p ≥ 1, the image Vϕ0 (M p (Rd )) is isometrically isomorphic
to the Bargmann-Fock space or simply Fock space
Z
p
2
F p (Cd ) := F : Cd → C entire
|F (z)|p e− 2 π|z| dA(z) < ∞ .
Cd
These function spaces are described in more detail in Paper C, where we explore
subaveraging properties in such spaces. From the Fock space-perspective, we compute upper bounds for the quantity
sup
f ∈M p (Rd )\{0}
∥Vϕ0 f · χΩ ∥pLp (R2d )
∥Vϕ0 f ∥pLp (R2d )
for general ν-porous sets Ω ⊆ R2d . For p ≥ 1, the above quantity serves as a
measure of optimal localization on Ω for signals in M p (Rd ) Notably, for p = 2, we
retrieve the operator norm of the Daubechies operator PΩϕ0 : L2 (Rd ) → L2 (Rd ).
14
1.3. The dynamical Hardy’s uncertainty principle
Theorem 1.2.8. (FUP for modulation spaces) Let 0 < h ≤ 1 be a continuous
parameter, and suppose that Ω(h) ⊆ R2d is an h-dependent family of sets which is
ν-porous on scales h to 1. Then for all p ≥ 1, there exist constants C, β > 0 only
dependent on ν (and d) such that
sup
f ∈M p (Rd )\{0}
∥Vϕ0 f · χΩ(h) ∥pLp (R2d )
∥Vϕ0 f ∥pLp (R2d )
≤ Chβ ∀ 0 < h ≤ 1.
(1.2.5)
Remark. In the above theorem, we again find explicit estimates for the exponent
β > 0 (see the remark at the end of Section C.3.1). Moreover, based on direct
estimates of the Nyquist density, we are able to reproduce the same upper bound
n
for the radial Cantor iterates as in Theorem 1.2.7 when πR2 ∼ M 2 . However, with
this approach alone, we are unable to evaluate the preciseness of our estimates.
Remark. Interestingly, our analog in Theorem 1.2.8 imposes milder conditions
on the porous sets compared to the original FUP in Theorem 1.2.5. In particular,
we do not require our h-dependent family of porous sets Ω(h) to be bounded in
terms of 0 < h ≤ 1. In addition, we may consider a fixed upper scale bound (equal
to 1 rather than h−1 ).
Utilizing a similar technique to that of Theorem 1.2.8, we formulate an FUP for
Guassian Gabor multipliers. Here we consider an h-dependent family of lattices
Λ(h) whose fundamental domain DΛ(h) all satisfy
DΛ(h) ⊆ BhL (0) ∀ 0 < h ≤ 1 for some constant L > 0.
(1.2.6)
Theorem 1.2.9. (FUP for Gabor multiplers) Let 0 < h ≤ 1 be a continuous
parameter, and suppose that Ω(h) ⊆ R2d is an h-dependent family of sets which is
ν-porous on scales h to 1. Consider the Gaussian Gabor multiplier
X
ϕ0
GΛ(h),Ω(h)
: L2 (Rd ) → L2 (Rd ), f 7→ |DΛ(h) |
Vϕ0 f (λ)π(λ)ϕ0 ,
λ∈ΛΩ (h)
where ΛΩ (h) := λ ∈ Λ |Ω(h) ∩ (λ + DΛ(h) )| > 0 ⊆ Λ(h). Suppose further that
all the lattices Λ(h) satisfy condition (1.2.6) with some universal constant L > 0.
Then there exists constants C, β > 0 only dependent on ν, L (and d) such that the
operator norm is bounded by
ϕ0
∥GΛ(h),Ω(h)
∥op ≤ Chβ ∀ 0 < h ≤ 1.
Remark. Although our analysis and results are primarily based on the Gaussian
window, the results of Theorem 1.2.8 and 1.2.9 can be extended to a larger class of
window functions. We refer the reader to Appendix C.D for a discussion on these
extensions, which include any finite linear combination of 1-dimensional Hermite
functions.
1.3
The dynamical Hardy’s uncertainty principle
The original Hardy’s uncertainty principle appears as a sharp decay estimate for
a function and its Fourier transform, first developed for unidimensional signals
15
Chapter 1. Preliminaries and motivation
in the seminal 1933-paper [56] and later extended to higher dimensions (see, e.g.,
[40]). Keeping the same normalization for the Fourier transform as in the previous
sections, the statement reads:
Theorem 1.3.1. (Hardy’s Uncertainty Principle) Suppose f ∈ L2 (Rd ) satisfies
the decay conditions
2
|f (x)| ≤ Ke−πα|x|
2
and |fˆ(ξ)| ≤ Ke−πβ|ξ|
for some constants α, β, K > 0.
(i) If αβ > 1, then f ≡ 0.
2
(ii) If αβ = 1, then f = ce−πα|x| for some c ∈ C.
The first analog in the joint representation was discovered by Gröchenig and
Zimmermann in [52] at the beginning of this century.
Theorem 1.3.2. Suppose (f, g) ∈ S × S ′ (Rd ) such that
2
|W (f, g)(x, ξ)| ≤ Ke−2π(α|x|
+β|ξ|2 )
for some constants α, β, K > 0.
(i) If αβ > 1, then W (f, g) ≡ 0 so f ≡ 0 or g ≡ 0.
(ii) If αβ = 1 and W (f, g) ̸≡ 0, then both f and g are multiples of the time2
2
frequency shifted Gaussian π(x0 , ξ0 )e−πα|x| = e2πiξ0 ·x eπα|x−x0 | for some
d
constants x0 , ξ0 ∈ R .
Another important contribution is the later work [11] by Bonami, Demange
and Jaming, which contains several estimates on the largest possible decay of the
Ambiguity function or equivalently the (cross-)Wigner distribution. Among these,
we draw focus to Corollary 6.5, which separates the decay condition in time and
frequency. From this corollary, we can deduce the following decay conditions:
Corollary 1.3.1. Suppose f, g ∈ L2 (Rd ) such that
|W (f, g)(x, ξ)| ≤ Ke−2π
P
j
αj |xj |2
and |W (f, g)(x, ξ)| ≤ Ke−2π
P
j
βj |ξj |2
for some constants αj , βj , K > 0. If for some j = 1, . . . , d the product αj βj > 1,
then W (f, g) ≡ 0 so f ≡ 0 or g ≡ 0.
Thus, this part of the thesis differs from the previous sections, in that, we are
no longer searching for an analog (or even new analogs) in the joint representation.
Rather, we showcase the benefit of a joint time-frequency perspective when working
with dynamical versions of Hardy’s uncertainty principle, regarding solutions of
the Schrödinger equation.
It turns out that the original Hardy’s uncertainty principle in Theorem 1.3.1
is equivalent to a sharp uniqueness result for the free Schrödinger equation
∂u
(t, x) = i∆u(t, x),
∂t
16
(1.3.1)
1.3. The dynamical Hardy’s uncertainty principle
where the solution cannot have strong decay at two distinct times. This dynamical
rephrasing was first considered in [30] and [15], and we reference the recent surveypaper [38] for an in-depth discussion and various proofs.
Theorem 1.3.3. (Theorem 3 in [38]) Let u ∈ C 1 ([0, T ], W 2,2 (Rd )) be the solution
of (1.3.1) such that for some constants α, β, K > 0
2
|u(0, x)| ≤ Ke−α|x|
2
and |u(T, x)| ≤ Ke−β|x| .
(i) If αβ > (4T )−2 , then u(t, x) ≡ 0.
2
(ii) If αβ = (4T )−2 , then u(0, x) = ce−(α+i/(4T ))|x| for some constant c ∈ C.
In the same spirit, we may also consider uniqueness properties for Schrödinger
equations with other, more general Hamiltonians. The Hamiltonian H = H(x, p),
in the position and momentum variables x, p ∈ Rd , describes the energy of a
physical system, where the free case refers to a system with no external potential
present. The associated Schrödinger equation is then given by
iℏ
∂u
b
(t, x) = Hu(t,
x),
∂t
b denotes the Weyl quantization of H (see Section D.2.1). Infamously,
where H
starting with [30], Escauriaza, Kenig, Ponce and Vega have in a series of papers
[31], [32], [33] studied Schrödinger evolutions for the free particle when perturbed by
certain time-dependent, bounded potentials V = V (t, x). Here, uniqueness results
are established based on the logarithmic-convexity properties of the solution. We
illustrate this with an example taken from [32] in the free case, where we try to
limit the number of technical details.
Example 1.3.1. (Log-convexity for free particle) Suppose u ∈ C 1 ([0, 1], W 2,2 (Rd ))
2
2
is the solution of (1.3.1) such that ∥u(0, x)eγ|x| ∥2 and ∥u(1, x)eγ|x| ∥2 are both
finite for some exponent γ > 0. We now define the function v := eϕ u for some
real-valued, smooth function ϕ = ϕ(t, x), and we shall evaluate the logarithmic
convexity properties of ∥v(t, ·)∥22 for a particular choice of ϕ. Specifically, for the
free Schrödinger equation, consider the function
ϕ(t, x) := γ|x + R t(1 − t)e1 |2 ,
where e1 = (1, 0, . . . , 0) denotes the unit vector in the first space coordinate and
R > 0 is some constant. This function is, in part, chosen so that ϕ(0, x) = ϕ(1, x) =
γ|x|2 . Moreover, we can show that
∂t2 ln ∥v(t, ·)∥22 ≥ −
R2
,
4γ
R2
which means ∥v(t, ·)∥2 e− 8γ t(1−t) is logarithmically-convex so that
2(1−t)
∥v(t, ·)∥22 ≤ ∥v(0, ·)∥2
R2
8γ t(1−t)
∥v(1, ·)∥2t
2 e
∀ t ∈ [0, 1].
17
Chapter 1. Preliminaries and motivation
Expressing v in terms of u and evaluating at t = 12 , the above inequality reads
Z
2
2
2
2
R
e2γ|x+ 4 e1 | |u(1/2, x)|2 dx ≤ ∥u(0, x)eγ|x| ∥2 ∥u(1, x)eγ|x| ∥2 eR /(32γ) .
Rd
In particular, for any 0 < ϵ < 1, we have that
Z
2
2
2
|u(1/2, x)|2 dx ≤ ∥u(0, x)eγ|x| ∥2 ∥u(1, x)eγ|x| ∥2 eR
1−4γ 2 (1−ϵ)2 /(32γ)
.
BϵR/4
Thus, whenever γ > 12 , we may choose ϵ > 0 sufficiently small such that the factor
2
2
2
eR (... ) converges to zero as R → ∞. Since ∥u(0, x)eγ|x| ∥2 and ∥u(1, x)eγ|x| ∥2
are both assumed to be finite, it follows that u(1/2, ·) ≡ 0 and therefore u ≡ 0.
Remark. Comparing the estimate for γ in Example 1.3.1 with Theorem 1.3.3, we
read that the estimate γ > 21 is not sharp, and the sharp estimate is, in fact, given
by γ > 14 . Unsurprisingly, for sharp(er) estimates and also if we include certain
bounded perturbations V (t, x) in the Hamiltonian, the log-convexity argument
becomes more technical.
Following the series [30]-[33] by Escauriaza, Kenig, Ponce and Vega, we have
the more recent contributions [13], [14] by Cassano and Fanelli on the Schrödinger
evolution of systems based on the harmonic oscillator and magnetic potentials,
including some bounded perturbations of these. Interestingly, their uniqueness
results rely directly on estimates obtained by Escauriaza, Kenig, Ponce and Vega,
where, by a change of variables, Cassano and Fanelli are able to reduce their
problem to a question regarding the (perturbed) free case.
In this thesis, however, we consider a different approach to the dynamical
Hardy’s uncertainty principle for a certain family of Hamiltonians. More precisely,
we shall consider real-quadratic Hamiltonians, i.e.,
H(x, p) =
d
X
aj,k xj pk for aj,k ∈ R,
j,k=1
where the harmonic oscillator and the uniform magnetic potentials are examples of
such systems. Our interest in this particular family stems from a remarkable result
regarding the Wigner distribution of the solution of the associated Schrödinger
equations. Namely, it turns out that the Wigner distribution of the solution u(t, x)
equals, up to a linear coordinate transformation SH = SH (t) : R2d → R2d , the
Wigner distribution of the initial condition u0 (x) := u(0, x), i.e.,
W u(x, t) = W u0 (SH (x, ξ)).
(1.3.2)
The details surrounding the coordinate transform SH are provided in Paper D,
where we discuss how SH is related to the symplectic mechanics associated with
real-quadratic Hamiltonians. With identity (1.3.2), we are able to establish uniqueness results for the solution u based directly on the analog Hardy’s uncertainty
principle for the Wigner distribution, specifically Corollary 1.3.1.
18
Chapter 2
Summary of papers
The research consists of four papers, presented in full-text in Part II. The first
three papers, Paper A, B and C, are all related to the fractal uncertainty principle
in the joint representation. The fourth and final paper, Paper D, is focused on
the dynamical Hardy’s uncertainty principle from the joint perspective. Below, a
summary of each paper is included. Any changes or additions to the published
versions are either outlined below or highlighted as footnotes in each paper. In
order to keep a more consistent style throughout, some of the notation has been
updated.
2.1
Paper A: Daubechies’ time-frequency
localization operator on Cantor type sets I
In the first paper [62], we consider Daubechies’ time-frequency localization operator with a Gaussian window and a radially symmetric weight. To begin with,
we present norm estimates for two introductory examples of localization on radially symmetric subsets. This is followed by an analysis of optimal localization on
the n-iterate radially symmetric mid-third Cantor set. In all the radial cases, our
norm estimates are predicated upon our explicit knowledge of the eigenvalues (and
eigenvectors) of the localization operator. Specifically, for the mid-third Cantor set
construction, we introduce the notion of relative areas, which we utilize to derive
precise asymptotes for the operator norm of the associated Daubechies’ operator.
Here we find that the operator norm decays exponentially to zero, even if we consider certain growth conditions on the outer radius of the n-iterate Cantor set so
that the measure of the sequence of subsets diverges. This result coincides with
the behaviour in the separate representation, and can therefore be viewed as a first
version analog of the fractal uncertainty principle in the joint representation.
19
Chapter 2. Summary of papers
2.2
Paper B: Daubechies’ time-frequency
localization operator on Cantor type sets II
The second paper [63] utilizes the machinery developed in the previous Paper A,
and naturally extends our localization result to localization on n-iterate radial
Cantor set with an arbitrary base M > 1 and alphabet A . In particular, we
estimate a common upper bound asymptote for the operator norm in terms of the
base M and the alphabet size |A | of the Cantor set. Although, this asymptote is
the overall optimal asymptote, considering all radial iterates with the same base
and with alphabets of the same size, a careful asymptotic analysis reveals that the
asymptote is never precise for every alphabet A with fixed size 0 < |A | < M .
We also consider the notion of indexed radial Cantor iterates, where the base
and alphabet are indexed by the iterations and may vary by each new iteration.
While some properties naturally transfer from the non-indexed to the indexed case,
we cannot always guarantee that the operator norm of the associated Daubechies’
operator decays to zero for increasing iterates. Concluding the paper, we present
sufficient conditions on the bases {Mj }j and alphabets {Aj }j so that we maintain
good control of the operator norm.
2.3
Paper C: A fractal uncertainty principle for
the short-time Fourier transform and Gabor
multipliers
In the third paper [64], we study a more general version of the fractal uncertainty
principle for the STFT with the Gaussian window, where we consider localization
on general ν-porous sets. In this case, we reformulate our localization problem
to an equivalent localization problem in the Bargmann-Fock space. In particular,
Daubechies’ localization operator is replaced by Toeplitz operators on the Fock
space. From this perspective, we are able to prove an analog fractal uncertainty
principle in the joint representation not only for signals in L2 (Rd ) but also for
signals in the modulation spaces M p (Rd ) for p ≥ 1. In our proof, we utilize
the subaveraging properties in the Fock space to construct an inductive scheme
with repeated, simple estimates of the maximal Nyquist density of the porous sets.
Specifically, for Cantor iterates, assuming certain growth rates of the initial interval
or initial radius, we can estimate the Nyquist density directly. Interestingly, for
radial Cantor iterates, this estimate coincides with the asymptote of Paper B.
Finally, we translate our results to discrete Gaussian Gabor multipliers.
Additions to published version
• We have included an additional appendix, Appendix C.D, which is not part
of the original published version. In this appendix, we identify the central
property from the Fock space that we needed in our proof, namely the subaveraging property. Based on this observation, we discuss extensions and
20
Chapter 2. Summary of papers
limitations of the technique in Paper C to STFTs with other window functions. In particular, for unidimensional signals, we show that the fractal
uncertainty principle holds in the joint representation when the window is
any finite linear combination of Hermite functions.
2.4
Paper D: Notes on Hardy’s uncertainty
principle for the Wigner distribution and
Schrödinger evolutions
In the final paper [65], we consider the dynamical Hardy’s uncertainty principle
for Schrödinger equations with real-quadratic Hamiltonians. For such Schrödinger
equations, we know that the Wigner distribution of the solution equals the Wigner
distribution of the initial condition, up to a linear coordinate transformation. By
combining this fact with the analog Hardy’s uncertainty principle for the Wigner
distribution, we establish a general uniqueness result, where the solution cannot
have strong decay at two distinct times. Specifically, this approach reproduces
known, sharp Hardy type estimates for the free Schrödinger equation, harmonic
oscillator and uniform magnetic potentials. Furthermore, based on Williamson’s
diagonalization theorem, we present a scheme for systems based on real-symmetric
positive definite matrices, from which we may extract explicit and also new Hardy
type estimates.
Additions and comments to published version
• We have included an appendix, Appendix D.A, which is not part of the
original publication, where we recap the proof of Williamsons’ theorem with
focus on the constructive nature of the proof.
• It is worth noting that in this paper we use a different normalization for the
Fourier transform than in the previous three papers. This normalization is
chosen in accordance with our Weyl quantization procedure for a physical
system. Naturally, we have also renormalized the Wigner distribution, this
time including the reduced Planck’s constant ℏ (for non-physical systems we
can think of ℏ = 1).
• So far we have reserved the bracket notation ⟨·, ·⟩ for inner products between
functions, where we have used the single dot notation for inner products
between euclidean vectors. However, in this paper, we oftentimes encounter
inner products between euclidean vectors which include some matrix. For
improved readability, we have therefore chosen in Paper D to denote these
inner products by
⟨M x, y⟩ rather than M x · y for x, y ∈ Rd and M is some d × d matrix.
21
Part II
Research Papers
Paper A
Daubechies’ time–frequency localization operator on
Cantor type sets I
Helge Knutsen
Published in
Journal of Fourier Analysis and Applications, 2020, volume 26, issue 3.
Paper A
Daubechies’ time-frequency
localization operator on Cantor
type sets I
Abstract
We study Daubechies’ time-frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window
and a spherically symmetric weight as this choice yields explicit formulas for
the eigenvalues, with the Hermite functions as the associated eigenfunctions.
Inspired by the fractal uncertainty principle in the separate time-frequency
representation, we define the n-iterate mid-third spherically symmetric Cantor set in the joint representation. For the n-iterate Cantor set, precise
asymptotic estimates for the operator norm are then derived up to a multiplicative constant.
A.1
Introduction
The problem of localizing signals in time and frequency is an old and important
one in signal analysis. In applications, we often wish to analyze signals on different
time-frequency domains, and we would therefore attempt to concentrate signals
on said domains. Different approaches for how to construct such time-frequency
localization operators have been suggested, either based on a separate or joint
time-frequency representation of the signal (see [79], [22]). The localization operators, regardless of which we choose to work with, will however be limited by the
fundamental barrier of time-frequency analysis, namely the uncertainty principles,
which state that a signal cannot be highly localized simultaneously in both time
and frequency. With regard to the localization operator, the limits posed by the
uncertainty principles translate into the associated operator norm, as it measures
the optimal efficiency of any given localization operator.
Many versions of the uncertainty principles exist (see [40]), and more recent
27
Paper A. Daubechies’ localization operator on Cantor type sets I
versions start to take into account the geometry of the time-frequency domains.
In particular, in [27], Dyatlov describes for signals in L2 (R) the development and
applications of a fractal uncertainty principle (FUP) in the separate time-frequency
representation, first introduced and developed in [29], [12], [28]. The relevant
localization operator is the standard composition of projections πT QΩ , where πT
and QΩ project onto the sets T in time and Ω in frequency, respectively. In the
context of the FUP, the sets T and Ω take the form of fractal sets. Here fractal
sets are defined in terms of the general notion of δ-regularity (see Def. 2.2. in
[27]), as families of sets T (h), Ω(h) ⊆ [0, 1], dependent on a continuous parameter
0 < h ≤ 1. The FUP is then formulated for this general class of sets when h → 0.
An illustrative example featured in [27] is the mid-third Cantor set, where both
the time and frequency domain can be regarded as h-neighbourhoods of the Cantor
set, say C (h). Notice that the FUP is originally framed such that the parameter
h is also encoded in the Fourier transform Fh (not unlike the normalization with
Planck’s constant in quantum mechanics) such that Fh f (ω) = √1h fˆ(ωh−1 ), where
fˆ denotes the Fourier transform of f . If we instead formulate the FUP
√ in terms
of the regular Fourier transform, we now consider the family C (h)/ h. Further,
if we translate the continuous h into discrete iterations n, we obtain a sequence
based on the n-iterate Cantor set, defined in increasing intervals depending on n.
More precisely, if T = Ω = Cn denotes the n-iterate defined in the interval [0, M ],
then the interval length satisfies
3n ∼ M 2 ,
(A.1.1)
√ n
which means |Cn | ∼ 2/ 3 → ∞ as n → ∞. However, by Theorem 2.13 in [27],
there exist constants α, β > 0 such that
∥πCn QCn ∥op ≤ αe−βn ∀ n ≥ 0.
(A.1.2)
We should expect some analogous result to the FUP (see Itinerary page 1 in
[47]) when extending to the joint time-frequency representation. Inspired by the
Cantor set example in separate time and frequency, we derive a similar localization for the Cantor set in the joint representation. In particular, we consider
Daubechies’ localization operator, first introduced in [22], based on the Short-Time
Fourier tranform, with a spherically symmetric weight function and a Gaussian
window. The reason for these restrictions is that, as was shown in the aforementioned paper, we obtain explicit expressions for the eigenvalues of the localization
operator, with the Hermite functions as the associated eigenfunctions.
The remainder of the paper is organized as follows: In Section A.2 we provide
a more detailed introduction to the Daubechies operator (Section A.2.1–A.2.2), in
addition to some necessary results in the spherically symmetric context (Section
A.2.3). We also make clear what we mean by a spherically symmetric Cantor
set (Section A.2.4). New results are found in Section A.3 and A.4, which contains several estimates for Daubechies’ operator localizing on different spherically
symmetric sets.
In particular, Section A.3 contains two introductory examples of localization
on spherically symmetric subsets, namely localization on a ring and a set of infinite
28
A.2. Preliminaries
measure. In Section A.4, we finally consider the n-iterate spherically symmetric
Cantor set. Here we derive precise asymptotic estimates (up to a multiplicative
constant) for the operator norm (Theorem A.4.2). A particular case of this twoparameter result, in terms of the radius R and iterate n, can be formulated as an
estimate solely in terms of the parameter n or R. In the spherically symmetric
context, we consider the condition
2
(A.1.3)
3n ∼ πR2 ,
similar to condition (A.1.1). Hence, under the above condition, let Pn denote the
Daubechies operator localizing on the n-iterate spherically symmetric Cantor set
defined in the disk of radius R > 0. Then for some positive constants c1 ≤ c2 the
operator norm satisfies
n2
n2
2
2
c1
≤ ∥Pn ∥op ≤ c2
.
(A.1.4)
3
3
This result is analogous to knowing the exponential β > 0 in (A.1.2) precisely.
A.2
Preliminaries
A.2.1
Fourier and short–time Fourier transform
For a function f : R → C the Fourier transform evaluated at point ω ∈ R is given
by
Z
fˆ(ω) =
f (t)e−2πiωt dt.
R
If we interpret f as an amplitude signal depending on time, then its Fourier transform fˆ corresponds to a frequency representation of the signal. The pair (f, fˆ)
does not, however, offer a joint description with respect to both frequency and
time. For this purpose, we consider the Short–Time Fourier Transform (STFT)
(see Chapter 3 in [47]).
The STFT is often referred to as the ”windowed Fourier transform” as this
transform relies on an additional fixed, non-zero function, ϕ : R → C, known as
a window function. At point (x, ω) ∈ R × R the STFT of f with respect to the
window ϕ is then defined as
Z
Vϕ f (x, ω) =
f (t)ϕ(t − x)e−2πiωt dt.
R
The transformed signal now depends on both time x and frequency ω, and we refer
to the (x, ω)–domain R2 as the phase space or the time-frequency plane.
We will restrict our attention to signals and windows in L2 (R), which, by
Cauchy-Schwarz’ inequality, implies that Vϕ f (x, ω) is well-defined for all points
(x, ω) ∈ R2 . Such restrictions also produce the following orthogonality relation
⟨Vϕ1 f1 , Vϕ2 f2 ⟩L2 (R2 ) = ⟨f1 , f2 ⟩⟨ϕ1 , ϕ2 ⟩ ∀ f1 , f2 , ϕ1 , ϕ2 ∈ L2 (R).
(A.2.1)
29
Paper A. Daubechies’ localization operator on Cantor type sets I
Equipped with the standard L2 -norms, we deduce that the STFT is a bounded
linear map with target space L2 (R2 ). If the window ϕ is normalized, i.e. ∥ϕ∥2 = 1,
then the STFT becomes, in fact, an isometry onto some subspace of L2 (R2 ).
Further, by identity (A.2.1), the original signal f can be recovered from its
phase space representation. Take any γ ∈ L2 (R) such that ⟨γ, ϕ⟩ ̸= 0, then the
orthogonal projection of f onto any g ∈ L2 (R) is given by
ZZ
1
⟨f, g⟩ =
Vϕ f (x, ω)Vγ g(x, ω) dxdω.
⟨γ, ϕ⟩
R2
A canonical choice for γ is to set it equal to ϕ. Assuming ϕ is normalized, these
projections then read
ZZ
⟨f, g⟩ =
Vϕ f (x, ω)Vϕ g(x, ω) dxdω.
(A.2.2)
R2
Since any signal f ∈ L2 (R) is entirely determined by such inner products, the
right-hand side of formula (A.2.2) provides a complete recovery from the STFT.
A.2.2
Daubechies’ localization operator
One approach for how to construct operators that localize a signal f in both
time and frequency was suggested by Daubechies in [22]. These operators can be
summarized as modifying the STFT of f by multiplication of a weight function,
say S(x, ω), before recovering a time-dependent signal. The weight function aims
at enhancing certain features of the phase space while diminishing others. Based
on formula (A.2.2), we consider the sesquilinear functional PSϕ on the product
L2 (R) × L2 (R), defined by
ZZ
PSϕ (f, g) =
S(x, ω)Vϕ f (x, ω)Vϕ g(x, ω) dxdω.
R2
Assuming PSϕ is a bounded functional, Riesz’ representation theorem ensures the
existence of a bounded, linear operator PSϕ : L2 (R) → L2 (R) such that
PSϕ (f, g) = ⟨PSϕ f, g⟩.
The operator PSϕ is our sought after time-frequency localization operator, which
we will refer to as Daubechies’ localization operator. From the above defninition,
PSϕ is characterized by the choice of weight S and window function ϕ.
In particular, any real-valued, integrable weight S will produce self-adjoint,
compact operators PSϕ whose eigenfunctions form aPcomplete basis for the space
L2 (R). Furthermore, the eigenvalues {λk }k satisfy k |λk | ≤ ∥S∥1 , in addition to
|λk | ≤ ∥S∥∞ for all k.
A.2.3
Spherically symmetric weight
For an arbitrary weight S and window ϕ it remains a challenge to determine the
eigenvalues of Daubechies’ localization operator PSϕ . However, in [22], Daubechies
30
A.2. Preliminaries
narrows in her focus to operators with a normalized Gaussian window
2
ϕ(t) = 21/4 e−πt ,
(A.2.3)
and a spherically symmetric weight
S(x, ω) = S (r2 ), where r2 = x2 + ω 2 .
For such operators, the Hermite functions1
21/4
Hk (t) = √
k!
1
− √
2 π
k
2
eπt
dk −2πt2
(e
), k = 0, 1, 2, . . .
dtk
(A.2.4)
are shown to constitute the eigenfunctions. Further, explicit expressions for the
associated eigenvalues {λk }k are derived.
Theorem A.2.1. (Daubechies) Let PSϕ denote the localization operator with weight
S(x, ω) = S (r2 ) and window ϕ equal to the normalized Gaussian in (A.2.3). Then
the eigenvalues of PSϕ are given by
Z
∞
λk =
S
r rk
0
π
k!
e−r dr, for k = 0, 1, 2, . . . ,
such that
PSϕ Hk = λk Hk ,
where Hk denotes the k-th Hermite function.
Observe that the normalized Gaussian in (A.2.3) coincides with H0 in (A.2.4).
It was shown recently in [4] that (for each j) the Hermite functions are also eigenfunctions of any localization operator with window Hj and a spherically symmetric
weight. Nevertheless, we will always assume the window ϕ to be the normalized
Gaussian.
We will consider the case when S projects onto a spherically symmetric subset
E ⊆ R2 . This means S equals the characteristic
function of some subset E ⊆ R+ ,
i.e., S (r) = χE (r), such that E = (x, ω) ∈ R2 | x2 + ω 2 ∈ E . As a matter of
convenience, we will denote the associated Daubechies operator simply by PE . By
Theorem A.2.1, the eigenvalue corresponding to the k-th Hermite function is then
given by
Z
λk =
π·E
rk −r
e dr, for k = 0, 1, 2, . . . ,
k!
(A.2.5)
1 Due to the choice of normalization for the Fourier transform, both the Gaussian and the Hermite functions are normalized differently than in [22]. The normalization is chosen in accordance
with Folland[41]. If hk denotes the k-th Hermite function in [22], this relates to Hk in (A.2.4)
√
1/4
by Hk (t) = √2 k hk ( 2πt).
2 k!
31
Paper A. Daubechies’ localization operator on Cantor type sets I
where π · E := {t ∈ R+ | tπ −1 ∈ E}. Since the above integrands will appear
frequently, we define, for simplicity, the functions
fk (r) :=
rk −r
e , r ≥ 0, for k = 0, 1, 2, . . .
k!
In Section A.4 we require two basic properties of the integrands {fk }k (see
Appendix A.A for additional details), namely
fk (k − r) ≤ fk (k + r) ∀ r ∈ [0, k] for k = 1, 2, 3, . . .
(A.2.6)
and
Z
Z
|E|
fk (r)dr ≤
E
f0 (r)dr = 1 − e−|E| for k = 0, 1, 2, . . . ,
(A.2.7)
0
where E is some measurable subset of R+ .
A.2.4
Cantor set
The mid-third Cantor set based in the interval [0, R] is constructed as follows:
Start with the interval C0 (R) = [0, R]. Each n-iterate Cn (R) is the union of 2n
disjoint, closed intervals {Ijn }j . To obtain the next iterate Cn+1 (R) remove the
open middle-third interval in every interval Ijn . Such iterations yield a nested
sequence C0 ⊇ C1 ⊇ C2 ⊇ . . . The mid-third Cantor set C (R) on the interval [0, R]
is then defined as the intersection of all the n-iterates, i.e.,
C (R) =
∞
\
Cn (R).
n=0
For each n-iterate, we define a corresponding map GR,n : R → [0, 1] by
(
0, x ≤ 0,
1
GR,n (x) =
·
(A.2.8)
|Cn (R)|
Cn (R) ∩ [0, x] , x > 0 for n = 0, 1, 2, . . . ,
which we refer to as the n-iterate Cantor function. These functions will come into
play in the latter part of Section A.4, where we will utilize the fact that {GR,n }n
are all subadditive, i.e.,
GR,n (a + b) ≤ GR,n (a) + GR,n (b) ∀ a, b ∈ R,
(A.2.9)
which was shown by induction by Josef Doboš in [24].
In the spherically symmetric context, we consider the following Cantor set construction: For the disk of radius R > 0 centered at the orgin, we identify the
n-iterate with the subset
(A.2.10)
Cn (R) = (x, ω) ∈ R2 x2 + ω 2 ∈ Cn (R2 ) ⊆ R2 .
32
A.3. Examples of localization on spherically symmetric sets
This means we consider weights of the form
S (r) = χCn (R2 ) (r), for R > 0 and n = 0, 1, 2, . . .
Based on formula (A.2.5), the eigenvalues of PCn (R) can then be expressed as
Z
Z
λk (Cn (R)) =
fk (r)dr =
π·Cn (R2 )
A.3
fk (r)dr.
(A.2.11)
Cn (πR2 )
Examples of localization on spherically
symmetric sets
In this section we present estimates for the operator norm of Daubechies’ operator
localizing on two different spherically symmetric sets. For this purpose, it would
be sufficient to determine the largest eigenvalue of the operator and estimate said
eigenvalue. Nonetheless, even with identity (A.2.5), it may prove difficult to determine which eigenvalue is the largest. Under such circumstances, we will instead
attempt to derive a common upper bound for the eigenvalues.
A.3.1
Localization on a ring: Asymptotic estimate
The first example shows that any eigenvalue λk of Daubechies’ localization operator
can, in principle, be the largest eigenvalue. Consider localization on a ring of inner
radius R > 0 in phase space of measure 1, that is, the subset
E (R) := (x, ω) ∈ R2
π(x2 + ω 2 +) ∈ [πR2 , πR2 + 1]
(A.3.1)
with the associated localization operator PE (R) . By result (A.2.5), the eigenvalues
of PE (R) read
λk (R) := λk (E (R)) =
Z
πR2 +1
fk (r)dr for k = 0, 1, 2, . . .
πR2
Now, assume that πR2 ∈ [m, m + 1] for some m ∈ N ∪ {0}. Since the difference
fk (r) − fk+1 (r) is negative precisely when r > k + 1, we obtain the ordering
λ0 (R) ≤ λ1 (R) ≤ λ2 (R) ≤ · · · ≤ λm (R)
and
λm+1 (R) ≥ λm+2 (R) ≥ λm+3 (R) ≥ . . .
Under these conditions, either λm (R) or λm+1 (R) must be the largest eigenvalue.
In particular, if πR2 = m, then λm (R) is the largest eigenvalue. In the next
proposition we provide an estimate of the operator norm of PE (R) .
33
Paper A. Daubechies’ localization operator on Cantor type sets I
Proposition A.3.1. Let E (R) ⊆ R2 be as in (A.3.1). For any fixed πR2 ≥ 2,
there exists a positive, finite constant C such that the operator norm of PE (R) is
bounded by
1
1
√ R−1 − CR−3 ≤ ∥PE (R) ∥op ≤ √ R−1 + CR−3 .
π 2
π 2
Proof. Let n := ⌊πR2 ⌋, where ⌊·⌋ denotes the floor function, rounding down to
the nearest integer. Apply a max-min-approximation of the integrands fk (r) for
r ∈ [n, n + 2] ⊇ [πR2 , πR2 + 1]. In particular, fn (n) serves as an upper bound
and, by inequality (A.2.6), fn+1 (n) serves as a lower bound for the operator norm.
That is,
nn −n
nn+1 −n
e ≤ ∥PE (R) ∥op ≤
e .
(n + 1)!
n!
Once we combine this with Stirling’s approximation formula for the factorial
√
1 √
2π · nn+1/2 e−n ≤ n! ≤ e 12n 2π · nn+1/2 e−n for n = 1, 2, 3, . . . ,
we obtain
−1
1
1
1 −1/2
1
√ n
1+
e− 12n ≤ ∥PE (R) ∥op ≤ √ n−1/2 .
n
2π
2π
2
2
Expressing the above
√ inequality in terms of R, we use that πR − 1 ≤ n ≤ πR
−3
and factor out 1/ πR2 . The error terms ±CR , follows by Taylor expansion of
the remaining factors about 1/(πR2 ) = 0.
Remark. A careful reading of the Taylor series expansion reveals that for πR2 ≥ 2,
the inequalities in Proposition A.3.1 hold for constant C = π −2 .
A.3.2
Localization on set of infinite measure
Next, we consider a non-trivial example of localization on a spherically symmetric
set of infinite measure (see [75] for a similar example in the separate time-frequency
representation). Define the subset
n
E (s) := (x, ω) ∈ R2
π(x2 + ω 2 ) ∈
∞
[
o
[n, n + s] ,
(A.3.2)
n=0
which we can identify with an infinite number of equidistant intervals in R+ .
Although the above set has infinite measure, we maintain good control over the
operator norm of PE (s) , and we can produce precise estimates in terms of the
parameter s > 0.
Theorem A.3.1. Let E (s) ⊆ R2 be as in (A.3.2) with s ∈ [0, 1]. Then the operator
norm of PE (s) is bounded by
C(1 − e−s ) ≤ ∥PE (s) ∥op ≤ min{Cs, 1} ∀ s ∈ [0, 1] with C =
34
e
.
e−1
(A.3.3)
A.3. Examples of localization on spherically symmetric sets
Further, there exists s0 > 0 such that
∥PE (s) ∥op = C(1 − e−s ) ∀ 0 < s < s0 .
(A.3.4)
Proof. By formula (A.2.5), the eigenvalues read
λk (s) := λk (E (s)) =
Z
fk (r)dr =
S
n [n,n+s]
∞ Z
X
n=0
n+s
fk (r)dr for k = 0, 1, 2, . . .
n
For each integral over [n, n + s], consider the maximum of fk (r) for r ∈ [n, n + 1]
such that
λ0 (s) ≤ s
∞
X
f0 (n) = s
n=0
∞
X
s
= Cs
1 − e−1
(A.3.5)
for k = 1, 2, 3, . . .
(A.3.6)
e−n =
n=0
and
λk (s) ≤ s fk (k) +
∞
X
!
fk (n)
n=0
We now claim that the following inequality holds
fk (k) +
∞
X
fk (n) <
n=0
∞
X
f0 (n) for k = 1, 2, 3, . . .
(A.3.7)
n=0
For k = 1, inequality (A.3.7) is verified by computing the series explicitly. While
for k > 1, compare the series with the integral over R+ , that is
Z ∞
X
fk (n) ≤
fk (r)dr = 1.
n̸=k
0
Thus,
fk (k) +
∞
X
fk (n) ≤ 1 + 2fk (k) ≤ 1 + 2f2 (2) = 1 + 4e−2 for k = 2, 3, 4, . . .
n=0
Since 1+4e−2 < C, claim (A.3.7) follows. Combining results (A.3.5)–(A.3.7) yields
the upper bound in (A.3.3). In the lower bound case of (A.3.3), it is sufficient to
observe
∞ Z n+s
∞
X
X
λ0 (s) =
e−r dr = (1 − e−s )
e−n = (1 − e−s )C.
n=0
n
n=0
For the equality case (A.3.4), note that inequality (A.3.7) ensures that there exists
a constant 0 < C0 < C such that λk (s) ≤ C0 s for any k, s > 0. Since
(1 − e−s )s−1 → 1 from below as s → 0,
it follows that some s0 > 0 with property (A.3.4) exists.
35
Paper A. Daubechies’ localization operator on Cantor type sets I
Remark. In [7] Theorem 3, a more general localization result is presented for
signals f ∈ M p (R) with p ≥ 1. The result is similar as it provides an upper bound
when localizing on sparse sets in phase space. Applied to signals f ∈ M 2 = L2
and the subset E (s), Theorem 3 yields, in fact, the same upper bound.2
A.4
Localization on spherically symmetric
Cantor set
In this section we consider localization on the n-iterate spherically symmetric Cantor set, i.e., the set Cn (R) in (A.2.10). Hence, we consider the localization operator
PCn (R) and attempt to estimate its operator norm. The results are formulated in
Section A.4.1, with the proof strategy and formal proofs in the subsequent Sections
A.4.2–A.4.4.
A.4.1
Results: Bounds for the operator norm
Below two theorems regarding the operator norm of PCn (R) are presented. The
first theorem shows to what extent the operator norm is bounded by the first
eigenvalue λ0 (Cn (R)).
Theorem A.4.1. The operator norm of PCn (R) is bounded from above by
∥PCn (R) ∥op ≤ 2λ0 (Cn (R)) for n = 0, 1, 2, . . .
The second theorem is a precise asymptotic estimate of the operator norm of
PCn (R) (up to a multiplicative constant) based on the same asymptotic estimate
for λ0 (Cn (R)).
Theorem A.4.2. There exist positive, finite constants c1 ≤ c2 such that for each
n = 0, 1, 2, . . .
ln 2
ln
3
2πR2 + 1
· ∥PCn (R) ∥op ≤ c2 ∀ πR2 ∈ [0, 3n /2].
c1 ≤ n
2 1 − e−πR2 /3n
Proofs of Theorem A.4.1 and A.4.2 can be found in Section A.4.3 and A.4.4,
respectively. At the end of Section A.4.4, we also retrieve numerical estimates for
the constants in Theorem A.4.2.
ln 2
If we now enforce condition (A.1.3), and note that 2n ∼ (πR2 )2 ln 3 , we obtain
the following corollary:
Corollary A.4.1. Suppose that the radius R depends on the iterate n such that
n
πR2 ∼ 3 2 . Then there exists positive, finite constants c1 ≤ c2 such that
ln 2 −1
ln 2 −1
c1 πR2 ln 3 ≤ ∥PCn (R(n)) ∥op ≤ c2 (πR2 ln 3 .
2 In the original paper it is claimed that the upper bound from [7] is coarser than the result in
Theorem A.3.1. By closer inspection, this claim turns out to be incorrect, which was originally
pointed out in the published version of paper C.
36
A.4. Localization on spherically symmetric Cantor set
Note that the above corollary is the same as result (A.1.4), except that we
have expressed the inequality in terms of the radius R rather than the iterate
n. On this form we have been able to express bounds for the operator norm in
2
terms of quantity δ = ln
ln 3 , which is the fractal dimension of the mid-third Cantor
set. It would therefore be interesting to investigate whether the same or a similar
statement holds when localizing on different Cantor sets, with a different fractal
dimension 0 < δ < 1.
A.4.2
Main strategy: Relative areas
Both theorems are obtained from the integral formula (A.2.11) for the eigenvalues
{λk (Cn (R))}k . However, as the number of intervals in the n-iterate Cantor set
grows as 2n , it soon becomes rather impractical to evaluate these integrals directly.
Instead we consider the local effect on the integrals when increasing from one
iterate to the next. In particular, this means we initially consider the integral of
fk over a single interval, say [s, s + 3L] for s ≥ 0 and L > 0. Then we attempt
to determine the relative area left under the curve fk once the mid-third of the
interval is removed, i.e., we wish to understand the function
"Z
# "Z
#−1
Z
s+L
Ak (s, 3L) :=
s+3L
s
s+3L
fk (r)dr ·
fk (r)dr +
s+2L
fk (r)dr
.
(A.4.1)
s
Computing the above integrals, Ak (s, 3L) can alternatively be expressed
#
" k
X 1
n
−L
n
−2L
n
−3L
n
s − e (s + L) + e
(s + 2L) − e
(s + 3L)
Ak (s, 3L) =
n!
n=0
" k
#
−1
X 1
n
−3L
n
·
s −e
(s + 3L)
.
(A.4.2)
n!
n=0
Observe that Ak (s, 3L) is independent of the starting point s precisely when k = 0.
In particular,
1 + e−2L 1 − e−L
.
(A.4.3)
A0 (3L) := A0 (s, 3L) =
1 − e−3L
For this reason, calculations with regard to λ0 (Cn (R)) are significantly simpler
than for the remaining eigenvalues. In particular, we have the recursive relation
λ0 (Cn+1 (R)) = A0 (πR2 /3n )λ0 (Cn (R)) for n = 0, 1, 2, . . . ,
which in turn means
λ0 (Cn+1 (R)) = λ0 (C0 (R))
n
Y
A0 (πR2 /3j )
j=0
= 1 − e−πR
2
n
Y
A0 (πR2 /3j ).
(A.4.4)
j=0
37
Paper A. Daubechies’ localization operator on Cantor type sets I
Ideally, if all the relative areas Ak (s, L) were bounded by A0 (L) regardless of
starting point s > 0 and interval length L > 0, we would conclude that λ0 (Cn (R))
is always the largest eigenvalue. As it turns out, this is not the case, e.g.,
lim Ak (0, L) > lim A0 (L) for k = 2, 3, 4, . . .
L→0
L→0
Nonetheless, in Section A.4.3, we are able to determine a common bound for the
eigenvalues in terms of λ0 (Cn (R)). Here, Lemma A.4.4 is worth highlighting as
it relies on the subadditivity of the Cantor function. Next, in Section A.4.4, we
compute the asymptotic estimates for λ0 (Cn (R)).
A.4.3
Proof of Theorem A.4.1
We begin by comparing Ak (s, 3L) to A0 (3L) when s ≥ k.
Lemma A.4.3. Let {Ak }k be given by (A.4.1). Then
Ak (s, 3L) ≤ A0 (3L) ∀ s ≥ k, L > 0 and k = 0, 1, 2, . . .
Proof. Consider the derivative of Ak (s, L) with respect to s, which yields
"Z
#−2
s+3L
∂ Ak
(s, 3L) = Nk (s, L)
fk (r)dr
,
∂s
s
where
Nk (s, L) = fk (s + L) − fk (s + 2L)
Z
s+3L
fk (r)dr
s
Z
− fk (s) − fk (s + 3L)
s+2L
fk (r)dr.
s+L
By identity (A.4.2), it is clear that lims→∞ Ak (s, 3L) = A0 (3L) for all L > 0.
Thus, it suffices to show that Nk (s, L) ≥ 0 for all s ≥ k and L > 0 for k = 1, 2, 3, . . .
Introduce the function
h
i
Φk (r, s, L) := fk (r + s)fk (s + L) − fk (r + s + L)fk (s)
h
i
(A.4.5)
+ fk (r + s + L)fk (s + L) − fk (r + s)fk (s + 2L) .
Then we may express Nk (s, L) as a single integral over [0, L] such that
Z L
Nk (s, L) =
Φk (r, s, L) − Φk (r, s + L, L) dr.
0
Hence, the function Nk (s, L) is positive for all s ≥ k if the derivative of Φ(r, s, L)
with respect to s is negative. Consider each of the square bracket terms [. . . ] in
definition (A.4.5) separately, that is
Ψk (r, s, L, y) :=fk (r + s + y)fk (s + L)
−fk (r + s + L − y)fk (s + 2y) for y ∈ {0, L},
38
A.4. Localization on spherically symmetric Cantor set
so that Φk (r, s, L) = Ψk (r, s, L, 0) + Ψk (r, s, L, L).
In order to easily evaluate the derivative of Ψk , notice first that the arguments
of fk (·) in each term of Ψk sum to a fixed value, namely
(i) 2a := 2s + r + L + y.
Further, introduce the corrections to each argument
(ii) ϵ1 := 2−1 (L − r − y) and ϵ2 := 2−1 (L + r − 3y)
such that Ψk becomes
Ψk (. . . ) = fk (a − ϵ1 )fk (a + ϵ1 ) − fk (a − ϵ2 )fk (a + ϵ2 )
k
k i
e−2a h 2
a − ϵ21 − a2 − ϵ22
.
=
2
(k!)
Since a = a(s) is the only quantity in the above expression that depends s, we
obtain
k−1
∂Ψk
2e−2a h 2
a − ϵ21
ka + ϵ21 − a2
(. . . ) =
2
∂s
(k!)
(A.4.6)
i
2
2 k−1
2
2
− a − ϵ2
ka + ϵ2 − a
.
By (i)–(ii) and since r ≤ L, we always have the ordering
|ϵ1 (y)| ≤ |ϵ2 (y)| ≤ a(s, y) ∀ s ≥ k and y ∈ {0, L},
which means (A.4.6) is negative whenever the factor (ka + ϵ21 − a2 ) is negative.
The latest claim is easily verified as |ϵ1 | ≤ a − k, and therefore
ϵ21 − (a − k)2 = (ka − ϵ21 − a2 ) + k(a − k) ≤ 0.
Hence, for any y ∈ {0, L}, 0 ≤ r ≤ L and s ≥ k, we conclude that
∂Φk
∂Ψk
(. . . ) ≤ 0 =⇒
(. . . ) ≤ 0 =⇒ Nk (s, L) ≥ 0.
∂s
∂s
By the latest lemma, any shifted n-iterate Cantor set Cn (πR2 ) + s with s ≥ k
satisfies
Z
Z
2 n
fk (r)dr ≤ A0 (πR /3 )
fk (r)dr for n = 0, 1, 2, . . . ,
Cn+1 (πR2 )+s
Cn (πR2 )+s
which combined with (A.2.7) and then identity (A.4.4), yields
Z
Z
n
Y
fk (r)dr ≤
fk (r)dr
A0 (πR2 /3j )
Cn+1 (πR2 )+s
C0 (πR2 )+s
≤ λ0 (C0 (R))
j=0
n
Y
A0 (πR2 /3j ) = λ0 (Cn+1 (R)).
(A.4.7)
j=0
Next, we relate the integrals of fk over the shifted n-iterates to the integrals over
the non-shifted n-iterates.
39
Paper A. Daubechies’ localization operator on Cantor type sets I
Lemma A.4.4. Let L > 0. Then for every fixed k, n = 0, 1, 2, . . . , we have
Z
Z
(A)
fk (r)dr ≤
fk (r)dr and
Cn (L)∩[k,∞)
Cn (L)+k
Z
Z
fk (r)dr ≤
(B)
Cn (L)∩[0,k]
fk (r)dr.
Cn (L)+k
Proof. For case (A), since fk (r) is monotonically decreasing for r > k, it suffices
to verify
Cn (L) ∩ [k, r] ≤ (Cn (L) + k) ∩ [k, r] ∀ r ≥ k.
In terms of the Cantor function GL,n in (A.2.8), the above claim reads
GL,n (r) − GL,n (k) ≤ GL,n (r − k) ∀ r ≥ k,
which is the same subadditivity property as in (A.2.9).
For case (B), consider the reflection of elements Cn (L) ∩ [0, k] about the point
k, that is, consider the subset
Rn,k := r ≥ k | 2k − r ∈ Cn (L) ∩ [0, k] ,
(A.4.8)
By (A.2.6), we have that
Z
Z
fk (r)dr ≤
Cn (L)∩[0,k]
fk (r)dr.
Rn,k
Similarly to (A), in order to prove (B), it suffices to show that
Rn,k ∩ [k, r] ≤ (Cn (L) + k) ∩ [k, r] = L · GL,n (r − k) ∀ r ≥ k.
(A.4.9)
By definition (A.4.8), the set Rn,k satisfies
|Rn,k ∩ [k, r]| = |Cn (L) ∩ [2k − r, k]| = L GL,n (k) − GL,n (2k − r) .
We now apply the subadditivity of GL,n to GL,n (k) = GL,n (r − k) + (2k − r) ,
from which claim (A.4.9) follows.
Now, combine inequality (A.4.7) with Lemma A.4.4, to conclude
λk (Cn (R)) ≤ 2λ0 (Cn (R)) ∀ k, n ≥ 0,
which is a restatement of Theorem A.4.1.
A.4.4
Proof of Theorem A.4.2
We formulate a precise asymptotic estimate for the first eigenvalue.
40
A.4. Localization on spherically symmetric Cantor set
Proposition A.4.1. There exist positive, finite constants a1 ≤ a2 such that for
each n = 0, 1, 2, . . .
a1 ≤
2πR2 + 1
2
ln
ln 3
2n 1 − e−πR2 /3n
· λ0 (Cn (R)) ≤ a2 ∀ πR2 ∈ [0, 3n /2].
Proof. Combine the two identities (A.4.3), (A.4.4) to obtain
n Y
2
n
2
j
λ0 (Cn (R)) = 1 − e−πR /3
1 + e−2πR /3
for n = 0, 1, 2, . . .
j=1
By the above identity, it is sufficient to show that
n
ln 2 Y
ln
2
j
1
3
1 + e−2πR /3 ≤ a2 ∀ πR2 ∈ [0, 3n /2].
a1 ≤ 2πR2 + 1
2
j=1
Exchange the product for a sum, and the above inequality is equivalent to
ln a1 ≤
n
X
ln 1 + e
j=1
−x/3j
ln(x + 1)
ln 2 ≤ ln a2 ∀ x ∈ [0, 3n ].
− n−
ln 3
The above inequality can now be proven by means of the following two claims
(i) there exists a finite, positive constant β such that
0≤
∞ h
X
i
j
j
ln 1 + y 1/3 − y 1/3 ln 2 ≤ β for y ∈ [0, 1], and
j=1
(ii) there exist finite constants γ1 ≤ γ2 such that
γ1 ≤
n
X
j
ln(x + 1)
≤ γ2 for x ∈ [0, 3n ].
e−x/3 − n −
ln
3
j=1
For claim (i), consider the non-negative function g(y) := ln(1 + y) − y ln 2 for
y ∈ [0, 1]. Since |g ′ (y)| ≤ g ′ (0) = 1 − ln 2 for all y ∈ [0, 1], the function g(y) can be
bounded from above by the linear spline
(
y, y ∈ [0, 1/2]
h(y) := g ′ (0) ·
(1 − y), y ∈ [1/2, 1].
P∞
1/3j
Thus, the sum in claim (i) is bounded by
) for y ∈ [0, 1]. Since
j=1 h(y
g(0) = h(0) = 0, we may always assume that y > 0. Further, observe that for
j
any 0 < y ≤ 1, we have that y 1/3 ↗ 1 as j → ∞. In particular, for any fixed
41
Paper A. Daubechies’ localization operator on Cantor type sets I
j
0 < y ≤ 1, there exists a smallest j0 ∈ N such that y 1/3 ≥ 1/2 for all j ≥ j0 .
Based on our choice j0 , we split the sum


jX
jX
∞
∞
∞
0 −1
0 −1
X
X
X
j
j
j
j
j
h(y 1/3 ) =
h(y 1/3 ) +
h(y 1/3 ) = g ′ (0) 
y 1/3 +
(1 − y 1/3 ) ,
j=1
j=1
j=j0
j=0
j=j0
and consider each sum separately. While the first sum is possibly empty, in the
j0 −1
non-empty case, introduce the variable z1 := y 1/3
∈ [0, 1/2] such that
jX
0 −1
j
y 1/3 =
j=1
jX
0 −2
j
z13 ≤
j=0
∞
X
j
z13 ≤
j=0
∞
X
j
2−3 =: S1 .
(A.4.10)
j=0
j
Similarly for the second sum, introduce the variable z2 := y 1/30 ∈ [1/2, 1] such
that
∞
X
j
(1 − y 1/3 ) =
j=j0
∞
X
1/3j
(1 − z2
)≤
∞
X
j
(1 − 2−1/3 ) =: S2 .
(A.4.11)
j=0
j=0
j
By direct comparison with the geometric series, that is, 2−3 ≤ 2−j and 1 −
j
2−1/3 ≤ 3−j ln(2), both series S1 and S2 are convergent. Hence, claim (i) follows
with β = g ′ (0)(S1 + S2 ).
Pn
j
Claim (ii) is proven by similar means as (i), where we split the sum j=1 e−x/3 .
In particular, for a fixed x ∈ [0, 3n ], we split the sum at the point
j1 := max
j1
n
n
X
X
X
ln(x)
.
=
+
,0
such that
ln(3)
j=j +1
j=1
j=1
1
j1
If the first sum is non-empty, set z3 := e−x/3
j1
X
j
e−x/3 =
j=1
jX
1 −1
j=0
j
z33 ≤
∞
X
j=0
j
∈ [0, e−1 ] such that
z33 ≤
∞
X
j
e−3 =: S3 ,
(A.4.12)
j=0
which is a convergent series.3 For the second sum, we utilize for y ≥ 0 the inequalities 1 − y ≤ e−y ≤ 1 to obtain lower and upper estimates. By comparison with
the geometric series and since x/3j1 +1 ≤ 1, we conclude that
−
n
X
j
3
≤
e−x/3 − (n − j1 ) ≤ 0.
2 j=j +1
(A.4.13)
1
Finally, by combining the estimates of (A.4.12) and (A.4.13) with the bounds
ln(x+1)
ln(x+1)
ln(3) − 1 ≤ j1 ≤
ln(3) , claim (ii) follows with constants γ1 = −3/2 and
γ2 = 1 + S 3 .
3 In the original paper there is an error in the second equality where the summation parameter
j starts at 1 and ends at j1 , while it should start at 0 and end at j1 − 1. This error carries over
into the proceeding inequalities, but does not change the validity of the proof, only the numerical
estimate, which has been updated here.
42
A.4. Localization on spherically symmetric Cantor set
Now, by applying the estimates of Proposition A.4.1 with constants a1 ≤ a2 to
Theorem A.4.1, we obtain Theorem A.4.2 with constants c1 = a1 ≤ c2 = 2a2 .
Remark (Numerical estimates). From the proof of Proposition A.4.1, we are also
able to retrieve some numerical estimates for the constants a1 ≤ a2 . It should,
however, be noted that the method chosen in the proof is not meant to produce
optimal constants. Nevertheless, with S1 , S2 , S3 defined as in (A.4.10)–(A.4.12),
we obtain the estimates
3
ln a1 = − ln 2 ≈ −1.0397 and
2
ln a2 = (1 − ln 2)(S1 + S2 ) + (1 + S3 ) ln 2 ≈ 1.4263.
Acknowledgements
The author would like to thank Eugenia Malinnikova for many fruitful discussions
on the topic covered and for providing feedback on drafts of the manuscript. In
addition, the author would like to extend thanks to the anonymous referees for
their many constructive comments on the first draft.
43
Paper A. Daubechies’ localization operator on Cantor type sets I
44
Appendix
A.A
Omitted proofs in Section A.2.3
We shall prove the following two properties for the integrands {fk (r) :=
fk (k − r) ≤ fk (k + r) ∀ r ∈ [0, k] for k = 1, 2, 3, . . .
r k −r ∞
}k=0 :
k! e
(A.A.1)
and
Z
|E|
Z
f0 (r)dr = 1 − e−|E| for k = 0, 1, 2, . . . ,
fk (r)dr ≤
E
(A.A.2)
0
where E is some measurable subset of R+ .
Proof. (Property (A.A.1)) It is sufficient to show that the fraction
δk (r) :=
fk (k − r)
≤ 1 ∀ r ∈ [0, k].
fk (k + r)
By differentiation, δk′ (r) ≤ 0 for all r ∈ [0, k] and since δk (0) = 1, we are done.
Proof. (Property (A.A.2)) Since every fk is normalized, i.e., ∥fk ∥1 = 1, and fk (r)
is monotonically increasing for 0 < r < k and decreasing for r > k, we may assume
E to be an interval of finite measure. Define the function
Z L
Z s+L
gk (L, s) :=
f0 (r)dr −
fk (r)dr,
0
s
and note that it suffices to show that gk (L, s) ≥ 0 for all L, s ≥ 0 and every k.
Differentiating gk with respect to L,
e−s
∂gk
(s, L) = f0 (L) − fk (s + L) = e−L 1 −
(s + L)k ,
∂L
k!
√
reveals a single critical point at L = L0 := k es k! − s. By the second derivative
test, it follows that L = L0 represents a maximum for gk (L, s) with s > 0 fixed.
Since the other possible extrema occur when L = 0 or L → ∞, which both can
easily be verified to yield a non-negative gk (L, s), we conclude that gk (L, s) is
always non-negative.
45
Paper B
Daubechies’ time–frequency localization operator on
Cantor type sets II
Helge Knutsen
Published in
Journal of Functional Analysis, 2022, volume 282, issue 9.
Paper B
Daubechies’ time-frequency
localization operator on Cantor
type sets II
Abstract
We study a version of the fractal uncertainty principle in the joint timefrequency representation. Namely, we consider Daubechies’ localization operator projecting onto spherically symmetric n-iterate Cantor sets with an
arbitrary base M > 1 and alphabet A . We derive an upper bound asymptote up to a multiplicative constant for the operator norm in terms of the
base M and the alphabet size |A | of the Cantor set. For any fixed base and
alphabet size, we show that there are Cantor sets such that the asymptote
is optimal. In particular, the asymptote is precise for mid-third Cantor set,
which was studied in part I [62]. Nonetheless, this does not extend to every Cantor set as we provide examples where the optimal asymptote is not
achieved.
B.1
Introduction
There are many versions of the uncertainty principle, that all, in some form, state
that a signal cannot be highly localized in time and frequency simultaneously. One
recent version, described by Dyatlov in [27], referred to as the fractal uncertainty
principle (FUP) (first introduced and developed in [29], [12], [28]), states that a
signal cannot be concentrated near fractal sets in time and frequency. Fractal sets
are here defined broadly as families of time and frequency sets T (h), Ω(h) ⊆ [0, 1],
dependent on a continuous parameter 0 < h ≤ 1, that are so-called δ-regular with
constant CR ≥ 1 on scales h to 1 (see Def. 2.2. in [27] for details). The FUP is
then formulated for this general class when h → 0. While originally formulated in
the separate time-frequency representation, we search for an analogous result in
the joint representation as the uncertainty principles should be present regardless
of our choice of time-frequency representation (see Itinerary page 1 in [47]).
49
Paper B. Daubechies’ localization operator on Cantor type sets II
In the separate representation, Dyatlov considers the following compositions
of projections onto such time and frequency sets, χΩ(h) Fh χT (h) , where χE is the
characteristic function of a subset E ⊆ R, and Fh is a dilated Fourier transform
√ −1
Fh f (ω) = h F f (ωh−1 ). For signals f ∈ L2 (R), the FUP then states (see
Theorem 2.12 and 2.13 in [27]) that for a fixed 0 < δ < 1 there exists some
non-trivial exponent β > max{0, 1/2 − δ} such that
∥χΩ(h) Fh χT (h) ∥op = O (hβ ) as h → 0.
(B.1.1)
Although no further estimates for the exponent is provided in [27], Jin and Zhang
have made some progress in a recent paper [61]: For families of sets T (h), Ω(h) ⊆
[0, 1] that are δ-regular with constant CR ≥ 1 on scales h to 1, they have derived
an explicit estimate for β > 0 only dependent on 0 < δ < 1 and CR .
In the original statement, we might not immediately recognize (B.1.1) as an
uncertainty principle since the parameter h is also encoded in the Fourier transform. However, if we disentangle h from the Fourier transform,
(B.1.1) turns out
√
√
to be a statement regarding localization on the sets T (h)/ h in time
√and Ω(h)/√ h
in frequency. Depending on our choice of δ, the measures |T (h)/ h|, |Ω(h)/ h|
might, in fact, tend to infinity as h → 0.
One prominent subfamily of fractal sets, is the Cantor sets. This includes the
infamous mid-third Cantor set, but instead of subdividing into three pieces and
keeping two by each iteration, we generalize and subdivide into M > 1 pieces
labeled {0, 1, . . . , M − 1} and keep a fixed alphabet A of said pieces. With regard
to the Cantor set construction, we could just as well speak of iterations n → ∞
rather than an h-neighbourhood that tends to zero. For Cantor sets with base M ,
these two quantities are related by h ∼ M −n .
Moreover, if T (h) = Ω(h) correspond to an√n-iterate Cantor
set Cn (M, A )
√
based in [0, 1], their scaled counterparts T (h)/ h = Ω(h)/ h correspond to a
n
Cantor set based in [0, L] with L ∼ M 2 . Or equivalently, for such scaled n-iterate
Cantor sets, the intervals {Ij }j that make up n-iterate all satisfy
|Ij | ∼
n
1
∼ M− 2 .
L
(B.1.2)
This condition will be a point of reference in the subsequent discussion.
In the joint representation, we shall consider Daubechies’ localization operator,
based on the Short-Time Fourier Transform (STFT) with a Gaussian window, that
projects onto spherically symmetric subsets of the time-frequency plane. The nontrivial assumption of radial symmetry is effective as Daubechies’ operator has a
known eigenbasis, the Hermite functions, and explicit formulas for the eigenvalues.
With this powerful tool available, we estimate the operator norm when projecting
onto the family of generalized spherically symmetric Cantor sets with base M and
alphabet A defined in a disk of radius R > 0. Thus, with our approach, this
version of the FUP does not rely on the FUP in the separate representation, and
the result from the separate representation is used as inspiration rather than a
direct implication. Other treatments of localization on sparse sets utilizing the
STFT can be found in [39] and [7], where sparsity is described in terms of ”thin
at infinity” and the Nyquist density, respectively.
50
B.2. Preliminaries
The remainder of the paper is organized as follows: In Section B.2, we describe
Daubechies’ operator and the Cantor set construction in more detail. New results
are divided into two sections, B.3 and B.4: In the first section, we keep the base
M > 1 and alphabet A fixed throughout the iterations, while in the second, we
introduce the notion of an indexed Cantor set where the base and alphabet may
vary.
In Section B.3, we let the radius R = R(n) be dependent on the iterates n. For
base M , we initially consider the general case when
R(n) → ∞ as n → ∞ while πR2 (n) ≤ M n for n = 0, 1, 2, . . .
(B.1.3)
A special case of the above restriction, similar to condition (B.1.2) (and also (1.3)
in [62]), is
n
πR2 (n) ∼ M 2 .
(B.1.4)
Let Pn (M, A ) denote Daubechies’ localization operator that projects onto the niterate spherically symmetric Cantor set with base M and alphabet A defined in
a disk whose radius R(n) > 0 satisfies (B.1.4). Then for some constant B > 0, we
find that
n
|A | 2
∥Pn (M, A )∥op ≤ B
for n = 0, 1, 2, . . . ,
M
where |A | < M denotes the size of the alphabet. This is similar to the asymptote
for the mid-third Cantor set, which was estimated in part I [62] and was shown
to be precise. Compared to the general case, we always have that the asymptote
is precise for the alphabet A = A = {0, 1, . . . , |A | − 1} and never precise for
{M −1, M −2, . . . , M −|A |}. Moreover, for the alphabet A , the largest eigenvalue
corresponds the Gaussian eigenfunction, independent of radius and iterate, which
was not established for the mid-third Cantor set.
While several properties transfer naturally from the non-indexed to the indexed
case in Section B.4, for an arbitrary indexed base and alphabet, we do not always
maintain good control over the operator norm. Therefore we present some simple
conditions on the bases {Mj }j and alphabets {Aj }j which guarantee that the
operator norm decays to zero for increasing iterates n.
B.2
B.2.1
Preliminaries
Daubechies’ localization operator
In order to produce a joint time-frequency representation of a signal f ∈ L2 (R),
we consider the Short-Time Fourier Transform (STFT), dependent on a fixed,
non-zero window function ϕ : R → C. At point (x, ω) ∈ R × R, at time x and at
frequency ω, the STFT of f with respect to the window ϕ, is then given by
Z
Vϕ f (x, ω) =
f (t)ϕ(t − x)e−2πiωt dt.
R
51
Paper B. Daubechies’ localization operator on Cantor type sets II
If we assume ϕ ≡ 1, we retrieve the (regular) Fourier transform of f , without
any time-dependence. For a joint time-frequency description, we assume ϕ to be
non-constant. In particular, if we consider windows ϕ ∈ L2 (R), with ∥ϕ∥2 = 1, the
STFT becomes an isometry onto some subspace of L2 (R2 ), i.e., ⟨Vϕ f, Vϕ g⟩L2 (R2 ) =
⟨f, g⟩ for any f, g ∈ L2 (R). Thus, we obtain a weakly defined inversion formula,
where the original signal f is recovered from Vϕ f via inner products.
Daubechies’ localization operator, introduced in [22], is based on the idea of
modifying the STFT of f by a multiplicative weight function S(x, ω) before recovering a time-dependent signal. The purpose of the weight function is to enhance
and diminish different features of the (x, ω)-domain R2 , e.g., by projecting onto a
subset of R2 . Characterized by our choice of window ϕ and weight S, we denote
the localization operator by PSϕ , which can be weakly defined as
⟨PSϕ f, g⟩ := ⟨S · Vϕ f, Vϕ g⟩L2 (R2 ) ∀ f, g ∈ L2 (R).
Since its conception, this operator has not only been studied as an operator between L2 -spaces, but also more broadly as an operator between modulation spaces,
and therein questions regarding boundedness and properties of eigenfunctions
and egenvalues remain relevant (see [18], [20], [9]). If we stick to the L2 (R)context, and assume the weight to be real-valued and integrable, the operator
PSϕ : L2 (R) → L2 (R) becomes self-adjoint, compact. In particular, this means
that the eigenfunctions of PSϕ form an eigenbasis for L2 (R), and the operator
norm ∥PSϕ ∥op is given by the largest eigenvalue in absolute value.
Similarly to Daubechies’ classical paper [22] and what was done in part I [62],
we shall focus our attention to weights that are spherically symmetric, that is, for
some integrable function S : R+ → R, we consider
S(x, ω) = S (r2 ), where r2 = x2 + ω 2 .
(B.2.1)
Combined with a normalized Gaussian window,
2
ϕ(x) = 21/4 e−πx ,
(B.2.2)
the eigenfunctions of PSϕ are known, with explicit formulas for the associated eigenvalues:
Theorem B.2.1. (Daubechies [22]) Let the weight S and window ϕ be given by
(B.2.1) and (B.2.2), respectively. Then the eigenvalues of the localization operator
PSϕ read
Z
λk =
∞
S
0
r rk
π
k!
e−r dr for k = 0, 1, 2, . . . ,
such that PSϕ Hk = λk Hk , where Hk denotes the k-th Hermite function,
21/4
Hk (t) = √
k!
52
1
− √
2 π
k
2
eπt
dk −2πt2
(e
).
dtk
(B.2.3)
B.2. Preliminaries
Interestingly, it was recently shown in [4] that any Hermite function Hj as
window and spherically symmetric weight yield localization operators with the
same eigenbasis {Hk }k . Another area that is being investigated is the aptly named
inverse problem, where one instead derives properties of the weight (symbol) based
on knowledge of the eigenfuntions. E.g., for Daubechies’ operators with a Gaussian
window that project onto a simply connected domain D ⊆ R2 , we know by [1] that
if Hj is an eigenfunction for some j, then D reduces to a disk centered at the origin.
More general situations are studied in [3], [6].
Proceeding with the direct problem and Daubechies’ classical result, formula
(B.2.3) represents a powerful tool for analyzing the localization operator and estimating the operator norm. In the subsequent discussion, we will only consider operators on the form as in Theorem B.2.1. More precisely, we consider the case when
S projects onto some spherically symmetric subset E ⊆ R2 . That is, S (r) = χE (r)
for some subset E ⊆ R+ such that E = (x, ω) ∈ R2 x2 + ω 2 ∈ E . For the
sake of simplicity, we will denote the associated localization operator by PE , whose
eigenvalues are given by
Z
rk −r
e dr for k = 0, 1, 2, . . . ,
λk =
π·E k!
where π · E := t ∈ R+ tπ −1 ∈ E . Here it is worth noting that if we fix the
measure |E|, we optimize the operator norm if E corresponds to a ball centered
at the origin. More precisely,
Z
Z π|E|
rk −r
e dr ≤
e−r dr = 1 − e−π|E| for k = 0, 1, 2, . . . ,
(B.2.4)
π·E k!
0
which was proved in Appendix A in [62].
Since these will appear frequently, we shall denote the above integrands by
fk (r) :=
rk −r
e for k = 0, 1, 2, . . .
k!
We recognize the function fk (r) as a gamma probability distribution, which is
monotonically increasing for r ∈ [0, k] and decreasing for r ≥ k.
Finally, observe that these operators can also be studied from the perspective
of Toeplitz operators on the Fock space. In particular, in [42] Galbis considers
Toeplitz operators with radial symbols and derive non-trivial norm estimates for
such operators.
B.2.2
Generalized Cantor set construction
Fix a positive integer M , called the base, and a non-empty proper subset A ⊆
{0, 1, . . . , M − 1}, called the alphabet. The n-iterate (or n-order) discrete Cantor
set is then defined as a subset of ZM n = {0, . . . , M n − 1} of the form


n−1

X
aj M j aj ∈ A for j = 0, 1, . . . , n − 1 .
(B.2.5)
Cn(d) (M, A ) :=


j=0
53
Paper B. Daubechies’ localization operator on Cantor type sets II
The discrete version corresponds to the ”continuous” version based in the interval
[0, R] by
Cn (R, M, A ) = RM −n · Cn(d) (M, A ) + [0, RM −n ] for n = 0, 1, 2, . . . ,
(B.2.6)
−1
where a·X = {x | x·a ∈ X} and X +[0, b] = ∪x∈X [x, x+b] for set X and scalars
a, b > 0. Let |A | denote the cardinality of the alphabet A . Then the measure of
the n-iterate Cantor set is given by
n
|A |
|Cn (R, M, A )| =
R.
(B.2.7)
M
In particular, for base M = 3 and alphabet A = {0, 2}, we recognize (B.2.6)
as the standard mid-third n-iterate Cantor set, with measure (2/3)n R. Another
noteworthy alphabet, that will appear frequently in the subsequent discussion, is
A = A := {0, 1, . . . , |A | − 1},
which we will refer to as the canonical alphabet of size |A |. Such a redistribution
of the alphabet does not alter the fractal dimension, i.e., the sets Cn (R, M, A ) and
|
Cn (R, M, A ) will still share the fractal dimension lnln|A
M .
For each n-iterate based in [0, R], we define a corresponding map GR,n,M,A :
R → [0, 1], known as the Cantor function, given by
(
−1 0, x ≤ 0,
(B.2.8)
GR,n,M,A (x) := |Cn (R, M, A )|
|Cn (R, M, A ) ∩ [0, x]|, x > 0.
Since GR,n,M,A (x) = G1,n,M,A (xR−1 ), we set Gn,M,A := G1,n,M,A , for simplicity.
It is well known that for the mid-third Cantor set the associated Cantor function
is subadditive (see [24]). However, with an arbitrary alphabet, subadditivity can
no longer be guaranteed. Instead we present a weaker version, sufficient for our
purpose, utilizing the canonical alphabet (see Appendix B.A for details).
Lemma B.2.2. Let Gn,M,A denote the Cantor-function, defined in (B.2.8). Then
for any x ≤ y,
Gn,M,A (y) − Gn,M,A (x) ≤ Gn,M,A (y − x).
(B.2.9)
For A = A , the above inequality is just standard subadditivity.
For the disk of radius R > 0, centered at the origin, we consider a spherically
symmetric n-iterate, based on the n-iterate Cantor set in (B.2.6), as a subset of
the form
Cn (R, M, A ) = (x, ω) ∈ R2 x2 + ω 2 ∈ Cn (R2 , M, A ) ⊆ R2 .
(B.2.10)
This means we consider weights of the form
S (r) = χCn (R2 ,M,A ) (r) for R > 0 and n = 0, 1, 2, . . .
The eigenvalues of Daubechies’ localization operator PCn (R,M,A ) then read
Z
λk (Cn (R, M, A )) =
fk (r)dr for k = 0, 1, 2, . . .
(B.2.11)
Cn (πR2 ,M,A )
54
B.3. Localization on generalized spherically symmetric Cantor set
B.3
Localization on generalized spherically
symmetric Cantor set
In this section we describe the behaviour of the operator norm, ∥PCn (R,M,A ) ∥op
as a function of the iterates n. The results are formulated in Section B.3.1, with
proofs and proof strategy in the subsequent sections B.3.2-B.3.5.
B.3.1
Results: Bounds for the operator norm
Below we present three theorems regarding the operator norm of PCn (R,M,A ) .
The first theorem shows that ∥PCn (R,M,A ) ∥op can be bounded in terms of the
”first eigenvalue” λ0 (Cn (R, M, A )), thus revealing the significance of the canonical
alphabet A = {0, 1, . . . , |A | − 1}.
Theorem B.3.1. The operator norm of PCn (R,M,A ) is bounded from above by
∥PCn (R,M,A ) ∥op ≤ 2λ0 (Cn (R, M, A )).
Further, for the canonical alphabet A = A , we have that
∥PCn (R,M,A ) ∥op = λ0 (Cn (R, M, A )).
In the next theorem we present an upper bound estimate for the operator norm
∥PCn (R,M,A ) ∥op .
Theorem B.3.2. There exists a positive, finite constant B only dependent on |A |
and M such that for each n = 0, 1, 2, . . .
πR2 + 1
|
lnln|A
M
|A |n 1 − e−M −n πR2
· ∥PCn (R,M,A ) ∥op ≤ B
∀ πR2 ∈ [0, M n ].
Proofs of Theorem B.3.1 and B.3.2 are found in section B.3.3 and B.3.4, respectively.
Remark. If the alphabet, A , is equal the canonical alphabet, A , then the lefthand-side of the inequality of Theorem B.3.2 can be bounded from below by a
non-negative constant, thus making the asymptote precise.
If we now enforce condition (B.1.3) on the radius R, we obtain the following
result:
Theorem B.3.3. Suppose the radius R depends on the iterates n so that
R(n) → ∞ as n → ∞ while πR2 ≤ M n for all n = 0, 1, 2, . . .
Then there exist positive, finite constants BL ≤ BU only dependent on M and |A |
such that
55
Paper B. Daubechies’ localization operator on Cantor type sets II
(a) for an arbitrary alphabet A we have the upper bound
n
|
1− lnln|A
|A |
M
∥PCn (R(n),M,A ) ∥op ≤ BU
πR2 (n)
, and
M
(b) for the canonical alphabet A = A , we also have the lower bound
n
|
1− lnln|A
|A |
M
∥PCn (R(n),M,A ) ∥op ≥ BL
.
πR2 (n)
M
(c) Conversely, at any alphabet size 0 < |A | < M , there exist alphabets A such
that
∥PCn (R(n),M,A ) ∥op · ∥PCn (R(n),M,A ) ∥−1
op → 0 as n → ∞.
|
Recall that the quantity lnln|A
M is the fractal dimension of the Cantor set with
|
base M and alphabet A . It should be noted that the exponent lnln|A
M − 1 in
Theorem B.3.3 (a) was already suggested in [62]. Since the associated upper bound
holds for all alphabets and bases, this immediately begs the question whether
the asymptote is precise regardless of alphabet and base. By Theorem B.3.3 (b)
and also Corollary 4.1 in [62], we conclude that the asymptote is precise for the
Cantor set with canonical alphabet and for the mid-third Cantor set, respectively.
However, by Theorem B.3.3 (c), it becomes clear that we cannot extend this result
to every alphabet. A constructive proof of Theorem B.3.3 (c) is found in Section
B.3.5.
B.3.2
Main tool: Relative areas
Similarly to Section 4 in [62], our main tool is the concept of relative areas, namely
"
# "Z
#−1
s+T
X Z s+(a+1)T M −1
Ak,M,A (s, T ) :=
fk (r)dr ·
fk (r)dr
. (B.3.1)
s+aT M −1
a∈A
s
The relative areas measures the local effect on the integrals that define the eigenvalues λk (. . . ) when we increase from one iterate n to the next n + 1. These
are in general easier to work with rather than the eigenvalues themselves directly.
Hence, we shall attempt to derive properties of the relative areas that transfer to
the global behaviour of the eigenvalues.
Initially, note that A0,M,A (s, T ) is independent of the starting point s ≥ 0,
which yields the nice recursive relation for the first eigenvalue
λ0 (Cn+1 (R, M, A )) = A0,M,A (·, πR2 M −n )λ0 (Cn (R, M, A )).
(B.3.2)
For the canonical alphabet A = A , definition (B.3.1) reduces to
"Z
# "Z
#−1
−1
s+|A |T M
Ak,M,A (s, T ) =
56
s+T
fk (r)dr ·
s
fk (r)dr
s
,
(B.3.3)
B.3. Localization on generalized spherically symmetric Cantor set
from which the relative area A0,M,A (·, T ) attains the simple form
|A |
1 − e− M T
.
1 − e−T
A0,M,A (·, T ) =
(B.3.4)
This relative area will play a significant role throughout the subsequent discussion.
We conclude this section by showing A0,M,A (·, T ) to be monotone, and illustrate
how local effects can transfer to global behaviour.
Lemma B.3.4. The relative area A0,M,A (·, T ) is monotonically increasing in the
argument T .
Proof. Set ϵ :=
|A |
M
< 1, and simply differentiate (B.3.4) to obtain
∂
∂
A
(·, T ) =
∂T 0,M,A
∂T
1 − e−ϵT
1 − e−T
−2
= e−T ϵe(1−ϵ)T + (1 − ϵ)e−ϵT − 1 1 − e−T
.
It suffices to show that the second factor in the above expression is always positive,
i.e.,
hϵ (T ) := ϵe(1−ϵ)T + (1 − ϵ)e−ϵT − 1 ≥ 0 ∀ T ≥ 0.
The latest claim is evident as
∂hϵ
(T ) = ϵ(1 − ϵ)e−ϵT (eT − 1) ≥ 0 ∀ T ≥ 0 and hϵ (T ) → 0 as T → 0.
∂T
By monotonicity of A0,M,A (·, T ) and the the recursive relation (B.3.2), it follows that the associated eigenvalue λ0 (. . . ) is increasing as a function of the radius,
i.e.,
λ0 (Cn (R1 , M, A )) ≤ λ0 (Cn (R2 , M, A )) ∀ R1 ≤ R2 and n = 0, 1, 2, . . . (B.3.5)
Monotonicity will also prove particularly useful both in Section B.3.5 and Section
B.4.
B.3.3
Proof of Theorem B.3.1
To begin with, we compare the relative areas with the canonical alphabet, for
which we have the rather remarkable result.
Lemma B.3.5. Let {Ak,M,A }k be given by (B.3.3). Then for any 0 < |A | ≤ M
and s ≥ 0, T > 0, we have the ordering
Ak,M,A (s, T ) ≥ Ak+1,M,A (s, T ) for k = 0, 1, 2, . . .
57
Paper B. Daubechies’ localization operator on Cantor type sets II
Proof. For convenience, we define Ak (s, T, |A |T M −1 ) := Ak,M,A (s, T ), and show
that the difference
Ak (s, T, t) − Ak+1 (s, T, t) ≥ 0 ∀ t ∈ [0, T ].
Firstly, we write the difference with a common denominator, which, by Fubini’s
theorem, yields
"Z
s+t
Z
#−1
s+T
fk (y)dy
fk (x)dx
Z
−
s+t
Z
s+T
Z
=
s
#−1
s+T
fk+1 (y)dy
fk+1 (x)dx
s
s
s
s
"Z
s+t
fk (x)fk+1 (y) − fk (y)fk+1 (x) dydx
s
"Z
·
s+T
Z
fk (r)dr
s
#−1
s+T
fk+1 (r)dr
.
s
Since the integral
Z
s+t
Z
s
s+t
fk (x)fk+1 (y) − fk (y)fk+1 (x) dydx = 0,
s
the difference is reduced to
Ak (s, T, t) − Ak+1 (s, T, t)
Z s+t Z s+T h i−1
=
fk (x)fk+1 (y) − fk (y)fk+1 (x) dydx · . . .
.
s
s+t
Inserting the definition fk (r) :=
r k −r
k! e
into the above integrand, we obtain
fk (x)fk+1 (y) − fk (y)fk+1 (x) =
1
xk y k (y − x)e−(x+y) ,
k!(k + 1)!
which is always positive since s ≤ x ≤ s + t ≤ y ≤ s + T .
For the canonical alphabet, we can immediately conclude with the following
corollary:
Corollary B.3.1. The largest eigenvalue of Daubechies’ operator PCn (R,M,A ) is
λ0 (Cn (R, M, A )), and consequently the operator norm is given by
∥PCn (R,M,A ) ∥op = λ0 (Cn (R, M, A )).
Proof. Since the relative area A0,M,A (·, T ) is independent of the starting points
s ≥ 0 and bounds all {Ak,M,A (s, T )}k , it is clear that
λk (Cn+1 (R, M, A )) ≤ A0,M,A (·, πR2 M −n ) λk (Cn (R, M, A ))
58
B.3. Localization on generalized spherically symmetric Cantor set
which, by the relation (B.3.2) and observation (B.2.4), eventually yields
λk (Cn+1 (R, M, A )) ≤ λk (C0 (R, M, A ))
n
Y
A0,M,A (·, πR2 M −j )
j=0
≤ λ0 (Cn+1 (R, M, A )) for k = 1, 2, 3 . . .
For a general alphabet, we compare A0,M,A (·, T ) to Ak,M,A (s, T ) for starting
points s ≥ k.
Corollary B.3.2. Let {Ak,M,A }k be given by (B.3.1). Then for 0 < |A | ≤ M
A0,M,A (·, T ) ≥ Ak,M,A (s, T ) ∀ s ≥ k, T > 0 and k = 0, 1, 2, . . .
Proof. Since fk (r) is monotonically decreasing for r ≥ k, it follows that
Ak,M,A (s, T ) ≥ Ak,M,A (s, T ) for s ≥ k,
which combined with the ordering in Lemma B.3.5, yields the result.
By the same argument as in Corollary B.3.1, we utilize Corollary B.3.2 to
obtain the bound
Z
Z
fk (r)dr ≤ A0,M,A (·, πR2 M −n )
fk (r)dr
Cn+1 (πR2 ,M,A )+s
Cn (πR2 ,M,A )+s
≤ λ0 (Cn+1 (R, M, A )) for s ≥ k.
(B.3.6)
Relating the shifted iterates Cn (. . . ) + s to the non-shifted iterates Cn (. . . ), we
present an almost analogous statement to Lemma 3.5 in [62].
Lemma B.3.6. Let L > 0. Then for every fixed k, n = 0, 1, 2, . . . , we have
Z
Z
fk (r)dr and
(A)
fk (r)dr ≤
Cn (L,M,A )∩[k,∞[
Z
Cn (L,M,A )+k
Z
fk (r)dr ≤
(B)
Cn (L,M,A )∩[0,k]
fk (r)dr.
Cn (L,M,A )+k
Proof. Both cases (A) and (B) follow the same steps as in Lemma 3.5 in [62] but
with the subadditivity property for the mid-third Cantor function exchanged for
the weaker version of Lemma B.2.2 for the general Cantor function. Similarly to
[62], for the case (B), we use that fk (k + r) ≥ fk (k − r) for all r ∈ [0, k] and
reflect the elements Cn (L, M, A ) ∩ [0, k] about r = k before proceeding with weak
subadditivity.
Finally, by combining inequality (B.3.6) with Lemma B.3.6, we obtain the
desired result
λk (Cn (R, M, A )) ≤ 2λ0 (Cn (R, M, A )) for k, n = 0, 1, 2, . . .
59
Paper B. Daubechies’ localization operator on Cantor type sets II
B.3.4
Proof of Theorem B.3.2
Proceeding, we present an exact formula for the first eigenvalue:
Lemma B.3.7. The first eigenvalue of the operator PCn (R,M,A ) is given by
λ0 (Cn (R, M, A )) = 1 − e−M
−n
πR2
n
X
Y
e−aj M
−j
πR2
.
(B.3.7)
j=1 aj ∈A
Further, we have the inequality
λ0 (Cn (R, M, A )) ≤ 1 − e
−M −n πR2
n
Y
j=1
−j
1 − e−|A |M πR
1 − e−M −j πR2
2
!
,
(B.3.8)
with equality precisely when A = A = {0, 1, . . . , |A | − 1}.
Proof. For simplicity, set R := πR2 . By definition (B.3.1), it is straightforward
to compute the relative area A0,M,A , which inserted into the recursive relation
(B.3.2) yields
−(n+1)
λ0 (Cn+1 (R, M, A )) =
1 − e− R M
1 − e−RM −n
X
e−an+1 RM
−(n+1)
λ0 (Cn (R, M, A )).
an+1 ∈A
This in return means
λ0 (Cn+1 (R, M, A )) = 1 − e
Y
n+1
−R
j=1
−j
1 − e−RM
1 − e−RM −(j−1 )
!
X
e−aj RM
−j
.
aj ∈A
Q
−j −(j−1) −1
Since the product j 1 − e−RM (1 − e−RM
is telescoping, only the
initial denominator and final numerator remain, and identity (B.3.7) readily follows. For the inequality case, merely note that the negative exponential function is
monotonically decreasing and with the canonical alphabet A = {0, 1, . . . , |A |−1},
we recognize the appearance of the geometric series.
Using result (B.3.8) for the eigenvalue λ0 (Cn (R, M, A )), we compute the asymptotes of said eigenvalue, which, combined with Theorem B.3.1, concludes the proof
of Theorem B.3.2.
Proposition B.3.1. There exists a finite constant B ≥ 1 only dependent on |A |
and M such that
B −1 ≤
πR2 + 1
|
lnln|A
M
|A |n 1 − e−M −n πR2
· λ0 (Cn (R, M, A )) ≤ B ∀ πR2 ∈ [0, M n ].
Proof. The proof follows a similar technique to that of Proposition 4.1 in [62],
with minor modifications. By Lemma B.3.7, the first eigenvalue of the operator
60
B.3. Localization on generalized spherically symmetric Cantor set
PCn (R,M,A ) , with the canonical alphabet A , is given by
λ0 (Cn (R, M, A )) = 1 − e
−M −n πR2
n
Y
j=1
−j
1 − e−|A |M πR
1 − e−M −j πR2
2
!
.
Hence, it suffices to show
B
−1
≤ x+1
|
lnln|A
M
−n
|A |
n
Y
j=1
−j
1 − e−|A |M x
1 − e−M −j x
!
≤ B ∀ x ∈ [0, M n ].
Utilizing the factorization (1 − y k ) = (1 − y)(1 + y + y 2 + · · · + y k−1 ) for k ∈ N
and expressing the product as a sum, the above statement reads
n
X
j
j
j
− ln B ≤
ln 1 + e−x/M + e−2x/M + · · · + e−(|A |−1)x/M
j=1
ln(x + 1)
− n−
ln |A | ≤ ln B
ln M
∀ x ∈ [0, M n ].
These inequalities follow by the aid of two claims
(i) there exists a finite, positive constant β such that for y ∈ [0, 1]
∞ h i
X
j
j
j
j
−β ≤
ln 1 + y 1/M + y 2/M + · · · + y (|A |−1)/M − y 1/M ln |A | ≤ β,
j=1
and
(ii) there exists a finite positive constant γ such that
n
X
ln(x + 1)
−x/M j
≤ γ for x ∈ [0, M n ].
−γ ≤
e
− n−
ln
M
j=1
For claim (i), we consider the function ψk (y) := ln(1 + y + y 2 + · · · + y k−1 ) − y ln k
for y ∈ [0, 1]. Since ψk is continuous and smooth in [0, 1], we have that
ck := max
y∈[0,1]
dψk
(y) < ∞,
dy
from which we may define the linear spline
(
y, y ∈ [0, 1/2]
hk (y) := ck ·
(1 − y), y ∈ [1/2, 1].
By the fact that ψk (0) = ψk (1) = 0, it is clear that ψk is bounded by the spline
such that
−hk (y) ≤ ψk (y) ≤ hk (y) for y ∈ [0, 1].
P∞
j
Thus, the sum in claim (i) is bounded from above and below by ± j=0 h|A | (y 1/M ),
respectively. From here the proof is essentially the same as for claim (i) in Proposition 4.1 in [62]. The same goes for claim (ii), where the only modification is that
3j is exchanged for M j > 1.
61
Paper B. Daubechies’ localization operator on Cantor type sets II
B.3.5
Proof of Theorem B.3.3 (c): Counterexample to
precise asymptotic estimate
For our counterexample, we shall consider the reverse canonical alphabet, that is,
A = A := {M − 1, M − 2, . . . , M − |A |},
where M denotes the associated base of the Cantor set. As it turns out, this
Cantor set construction is merely a shifted version of the Cantor set with canonical
alphabet, A .
Lemma B.3.8. Let A denote the reverse canonical alphabet of size |A |. Then
the Cantor set with alphabet A and base M can be expressed as
n
X
Cn (R, M, A ) = R · M − |A |
M −j + Cn (R, M, A ) for n = 0, 1, 2, . . .
j=1
Proof. By definition (B.2.5), the discrete Cantor set with the reverse canonical
alphabet is given by


n−1

X
Cn(d) (M, A ) =
aj M j aj = M − |A |, . . . , M − 1 for j = 0, 1, . . . , n − 1


j=0


n−1

X
j
=
aj + M − |A | M
aj = 0, 1, . . . , |A | − 1 for j = 0, 1, . . . , n − 1


j=0
X
n−1
M j + Cn(d) (M, A ).
= M − |A |
j=0
The continuous version then follows once we apply definition (B.2.6) to the last
identity.
Hence, for the operator PCn (R,M,A ) , the eigenvalues read
Z
λk (Cn (R, M, A )) =
fk (r)dr for k = 0, 1, 2, . . . ,
2
πRinner
+Cn (πR2 ,M,A )
where the inner radius Rinner = Rinner (n, M, |A |) ≥ 0 is given by
n
X
2
Rinner
= R2 · M − |A |
M −j .
(B.3.9)
j=1
According to condition (B.1.3), we consider radii R(n) that tends to infinity as
n → ∞, which also means that the inner radius Rinner = Rinner (n) → ∞ as n → ∞.
Based on this inner radius, we may, in fact, exclude certain eigenvalues from being
the largest. More precisely, let ⌊·⌋ denote the floor function, rounding down to the
nearest integer. Since the difference between two integrands fk+1 (r)−fk (r) ≥ 0 for
2
r ≥ k + 1, it is clear that largest eigenvalue λk (. . . ) must have index k ≥ ⌊πRinner
⌋.
62
B.3. Localization on generalized spherically symmetric Cantor set
Proceeding, we consider the universal upper bound provided by Lemma B.3.6,
namely
Z
2
fk (r)dr ≥ λk (Cn (R, M, A )) for k = 0, 1, 2, . . . ,
(B.3.10)
k+Cn (πR2 ,M,A )
which holds for all alphabets, including A = A . We begin by computing the
associated relative areas as k → ∞.
Lemma B.3.9. Let Ak,M,A (s, T ) be the relative area given by (B.3.3) over the
interval [s, s + T ]. Suppose the starting point s depends on k so that s = ak for
a > 1. Then
h
i
|
1
1 − exp − |A
T
1
−
M
a
.
lim A
(ak, T ) =
(B.3.11)
k→∞ k,M,A
1 − exp −T 1 − a1
For s = k, the limit reduces to limk→∞ Ak,M,A (k, T ) =
|A |
M .
Proof. Set t := |A |T M −1 ∈ [0, T ] for simplicity. By inserting the definition of the
k
integrands fk (r) = rk! e−r , we obtain
lim Ak,M,A (ak, T ) = lim
k→∞
k→∞
fk (r)dr
s
t
−r
= lim
#−1
s+T
fk (r)dr
s
Z
k→∞
"Z
s+t
Z
e
0
r k
dr
1+
ak
"Z
T
e
−r
0
r k
dr
1+
ak
#−1
.
k
Now, identity limk→∞ 1 + xk = ex ensures we can utilize the dominated convergence theorem and exchange the order of the integrals and limit, so that
"Z
#−1
T
r k
r k
−r
lim A
(ak, T ) =
e
lim 1 +
e
lim 1 +
dr
dr
k→∞ k,M,A
k→∞
k→∞
ak
ak
0
0
"Z
#−1
Z t
T
1
1
−r (1− a
−r (1− a
)
)
dr
e
dr
=
e
Z
t
−r
0
0
)
1 − e−t(
for a > k.
1
1 − e−T (1− a )
1
1− a
=
For s = k, i.e., a = 1, the final integrands reduce to e−r(1− a ) = 1.
1
Remark. Comparing the above limit to the relative areas of λ0 (Cn (R, M, A )) in
(B.3.4), we find that limk Ak,M,A (ak, T ) = A0,M,A (·, (1 − a1 )T ).
Although the limit in Lemma B.3.9 has a familiar form, it must be handled
with some care as it, in fact, represents a lower bound rather than an upper one.
63
Paper B. Daubechies’ localization operator on Cantor type sets II
Lemma B.3.10. Suppose the factor a ≥ 1.
(A) The limit relative area is an infimum in the sense that
Ak,M,A (ak, T ) ≥ lim Aj,M,A (aj, T ) for k = 0, 1, 2, . . .
j→∞
(B) Conversely, fix ϵ > 0. Then
Ak,M,A (ak, T ) ≤ lim Aj,M,A ((1 + ϵ)j, T ) for ak + T ≤ (1 + ϵ)k.
j→∞
In particular, the inequality holds for T ≤
h
ϵi
ϵk
and a ∈ 1, 1 + .
2
2
Proof. We start with (B) and relate this to (A). Set t := |A |T M −1 and notice
that
Z ak+t
1
lim Aj,M,A ((1 + ϵ)j, T ) =
e−x(1− 1+ϵ ) dx
j→∞
ak
"Z
ak+T
·
#−1
e−y(
) dy
1
1− 1+ϵ
for k = 0, 1, 2, . . .
ak
On this form we compute the difference limj Aj,M,A ((1 + ϵ)j, T ) − Ak,M,A (ak, T ),
which, by the same technique as in Lemma B.3.5, yields
Z ak+t Z ak+T
i
h
y
x
e−(x+y) y k e 1+ϵ − xk e 1+ϵ dydx
ak
ak+t
"Z
ak+T
·
k −y
y e
ak
Z
#−1
ak+T
dy
e
1
−(1− 1+ϵ
)x
dx
.
ak
If the above integrand is always positive or always negative, this translates directly
to the sign of the difference. This is the same as asking if the function
y
x
ln Q(x, y) = k ln y −
− k ln x +
=: ln q(y) − ln q(x)
1+ϵ
1+ϵ
is always positive or always negative for ak ≤ x ≤ ak + t ≤ y ≤ ak + T . Since
′
the derivative (ln q) (x) ≥ 0 for x ∈ [k, (1 + ϵ)k], it follows that ln Q(x, y) ≥ 0 for
k ≤ x ≤ y ≤ (1+ϵ)k and thus part (B). For part (A), we simply consider 1+ϵ = a,
which yields (ln q)′ (x) ≤ 0 for x ≥ ak.
Now, it is tempting to suggest a common upper bound regardless of starting
point s = ak:
Example B.3.1. Suppose L > 1 is constant and that a ∈ [1, L] and T ≤ Lk for
all iterates n. By Lemma B.3.10 part (B), we have that
Ak,M,A (ak, T ) ≤ lim Aj,M,A (2Lj, T ) = A0,M,A ·, 1 − (2L)−1 T .
j→∞
However, this estimate for the relative area is essentiallyp
the same as for the eigenvalue λ0 (Cn (R, M, A )), only with a scaled radius, R 7→ 1 − (2L)−1 R, which, by
Theorem B.3.3 (a) and (b), does not change the asymptotes.
64
B.3. Localization on generalized spherically symmetric Cantor set
R∞
With this example in mind, we divide the integral k fk (r)dr into a significant
and insignificant part, thus, gaining more control over the starting points s = ak.
In the next lemma, we show what we mean by insignificant.
Lemma B.3.11. For any fixed ϵ, δ > 0, there exists a positive integer K, so that
the integral
Z
∞
fk (r)dr = e−(1+ϵ)k
(1+ϵ)k
k
X
n
1
(1 + ϵ)k < δ ∀ k ≥ K.
n!
n=0
Proof. Using the lower bound version of Stirling’s approximation formula for the
factorial
√
1
2πnn+ 2 e−n ≤ n! for n = 1, 2, 3, . . . ,
we determine a simplified upper bound for the summand
n n
k
k X
n
e
1
1 X (1 + ϵ)k
k
√ ≤ √ (1 + ϵ)k ek .
(1 + ϵ)k ≤ √
n!
n
n
2π n=1
2π
n=1
In the final inequality, we have used that the function
n
A
is monotonically increasing for 1 ≤ n ≤ Ae−1 .
gA (n) :=
n
Applying the negative exponential e−(1+ϵ)k to the upper bound, we obtain
e−(1+ϵ)k
k
X
n
1
k
(1 + ϵ)k ≤ √ exp − ϵ − ln(1 + ϵ) k ,
n!
2π
n=1
and since ϵ − ln(1 + ϵ) > 0 for any fixed ϵ > 0, we are done.
Proposition B.3.2. Suppose the radius R depends on the iterates n such that
R(n) → ∞ as n → ∞, and let γ > 0 be a multiplicative constant. Then for every
ϵ, δ > 0, there exists a positive integer N such that for every iterate n ≥ N and
every k ≥ γ · πR2 (n), we have that
Z
√
(A)
fk (r)dr ≤ 2λ0 (Cn ( ϵR, M, A )), and
Cn (πR2 ,M,A )+k ∩ k, 1+ 2ϵ k
Z
(B)
Cn (πR2 ,M,A )+k ∩
fk (r)dr ≤ δ · λ0 (Cn (R, M, A )).
1+ 2ϵ k,∞
Proof. We consider iterations n so that the relevant indices k ≥ γ · πR2 (n) ≥ 1.
For case (A), consider any fixed iterate n0 so that
M −n0 ≤
γ·ϵ
.
2
65
Paper B. Daubechies’ localization operator on Cantor type sets II
The interval size |I| of the n0 -iterate Cantor set Cn0 (πR2 (n), M, A ) then satisfies
ϵ
|I| = πR2 (n)M −n0 ≤ k ∀ k ≥ γ · πR2 (n).
2
By Lemma B.3.10 (B) and identity limk Ak,M,A (ak, T ) = A0,M,A (·, (1 − a−1 )T ),
we obtain
Z
fk (r)dr
Cn (... )+k ∩[k,(1+ 2ϵ )k]
(B.3.12)
Z ∞
n−1
Y
ϵ
2
−j
fk (r)dr ·
≤
A0,M,A ·,
πR M
.
1+ϵ
k
j=n
0
By the recursive relation (B.3.2) for λ0 (Cn (. . . )), it is clear that
ϵ
Z 1+ϵ
n−1
πR2 M −n0
Y
ϵ
2
−j
πR M
f0 (r)dr ·
A0,M,A ·,
1+ϵ
(B.3.13)
0
j=n0
√
≤ λ0 (Cn ( ϵR, M, A )),
p
√
where we have simplified the argument ϵ(1 + ϵ)−1 R ≤ ϵR by monotonicity of
λ0 (Cn (. . . )). Since the radius R(n) → ∞ as n → ∞, there exists a positive integer
n1 so that the integral
ϵ
Z 1+ϵ
Z ∞
πR2 (n)M −n0
2
f0 (r)dr ≥ 1 ≥
fk (r)dr ∀ n ≥ n1 .
(B.3.14)
0
k
Combining the three inequalities (B.3.12)–(B.3.14), then yields (A) with any N ≥
max{n0 , n1 }.
For case (B), observe first, by the subadditivity of the Cantor function Gn,M,A ,
that
Z
Z
fk (r)dr ≤
fk (r)dr.
Cn (... )+k ∩[(1+ 2ϵ )k,∞)
Cn (... )+(1+ 2ϵ )k
Further, by Corollary B.3.2 and the recursive relation (B.3.2) for λ0 (Cn (. . . )), we
have
Z ∞
Z
n−1
Y
fk (r)dr ≤
fk (r)dr
A0,M,A (·, πR2 M −j )
ϵ
ϵ
Cn (... )+(1+ 2 )k
(1+ 2 )k
j=0
"Z
#−1
Z ∞
πR2
=
fk (r)dr ·
f0 (r)dr
· λ0 (Cn (R, M, A )).
0
(1+ 2ϵ )k
We now apply Lemma B.3.11, whereas for any δ > 0 there exists a positive integer
n2 so that
Z ∞
Z πR2 (n)
δ
1
fk (r)dr <
and
f0 (r)dr ≥
∀ k ≥ γ · πR2 (n) and n ≥ n2 ,
ϵ
2
2
0
(1+ 2 )k
from which (B) follows with any N ≥ n2 . For both cases, we simply chose the
threshold N ≥ max{n0 , n1 , n2 }.
66
B.4. Further generalizations: Indexed Cantor set
Proposition B.3.3. Let A and A denote the canonical and reverse canonical
alphabet, respectively. Suppose the radius R depends on the iterates n such that
R(n) → ∞ as n → ∞ while πR2 (n) ≤ M n . Then the quotient
∥PCn (R(n),M,A ) ∥op · ∥PCn (R(n),M,A ) ∥−1
op → 0 as n → ∞.
Proof. Utilizing the universal upper bound (B.3.10), we have that
Z
2
sup
∥PCn (R,M,A ) ∥op ≤
fk (r)dr for n = 0, 1, 2, . . .
2
(n)⌋
k≥⌊πRinner
Cn (πR2 ,M,A )+k
Now fix some ϵ, δ > 0. Since Rinner (n) → ∞ as n → ∞, we can apply Proposition
B.3.2 (A) and (B), and conclude that for some threshold iterate N0
√
∥PCn (R,M,A ) ∥op ≤ 4λ0 (Cn ( ϵR, M, A ) + 2δ · λ0 (Cn (R, M, A )) ∀ n ≥ N0 .
To see why this inequality readily yields the desired result, we insert the estimates of Theorem B.3.3 (a) and (b). In particular, there exist a constant B > 0
(independent of ϵ) and threshold iterate N1 so that
n
|
1− lnln|A
|A |
−1
M
λ0 (Cn (R(n), M, A )) ≥ B
πR2 (n)
and
M
n
|
1− lnln|A
√
ln |A |
|A |
M
ϵ1− ln M πR2 (n)
λ0 (Cn ( ϵR(n), M, A )) ≤ B
∀ n ≥ N1 .
M
From these two estimates, and the fact that ∥PCn (R(n),M,A ) ∥op = λ0 (Cn (R, M, A )),
we obtain
ln |A |
2
1− ln M
∥PCn (R(n),M,A ) ∥op · ∥PCn (R(n),M,A ) ∥−1
+ 2δ ∀ n ≥ max{N0 , N1 }.
op ≤ B 4ϵ
Since ϵ, δ are arbitrary, and the constant B does not depend on either, we are
done.
B.4
Further generalizations: Indexed Cantor set
So far in our discussion the base and alphabet have been fixed throughout the
iterations. Suppose we now instead let these two quantities vary, to obtain an
even more general Cantor set construction. In particular, if we index the bases
∞
{Mj }∞
j=1 and the alphabets {Aj }j=1 according to each iteration, we obtain the
discrete construction


j−1
n
X

Y
Cn(d) ({Mj }j , {Aj }j ) :=
aj
Ml aj ∈ Aj for j = 1, 2, . . . , n , (B.4.1)


j=1
l=1
which in return yields the continuous version
−1 (d)
Cn (R, {Mj }j , {Aj }j ) =R M1 M2 . . . Mn
· Cn ({Mj }j , {Aj }j )
−1 + 0, R M1 M2 . . . Mn
for n = 0, 1, 2, . . . ,
(B.4.2)
67
Paper B. Daubechies’ localization operator on Cantor type sets II
where an empty product is, by convention, defined as 1. We shall refer to the
above construction, Cn (R, {Mj }j , {Aj }j ), as an indexed Cantor set.
Similarly to (B.2.10), we can also define an n-iterate spherically symmetric, indexed Cantor set, Cn (R, {Mj }j , {Aj }j ), and hence define the localization operator
PCn (R,{Mj }j ,{Aj }j ) . With condition (B.1.4) in mind, a natural restriction on the
radius seems to be
πR2 (n) ≤ γ · (M1 M2 . . . Mn )1/2 ,
(B.4.3)
for some finite constant γ > 0, and it is this restriction we utilize when formulating
our localization results. The results are found in section B.4.1, with proofs in
section B.4.2 and B.4.3.
B.4.1
Results: Sufficient decay conditions
As should be expected, for an arbitrary indexed Cantor set, we cannot guarantee
that the operator norm of the associated localization operator decays exponentially
or even converges to zero for increasing iterates. Below we present two theorems
that reflect this. In the first theorem we present some sufficient conditions for
which the operator norm indeed decays exponentially.
Theorem B.4.1. Suppose we have an indexed Cantor set Cn (R, {Mj }j , {A }j )
such that
|Aj |
≤ ϵ < 1 ∀ j ∈ N and Mj ∈ [M, M 1+δ ] for some finite δ > 0 and M > 1.
Mj
Further, suppose that the radius R satisfies the condition
1/2
πR2 (n) ≤ γ · M1 M2 . . . Mn
for some finite γ > 0. Then there exist finite constants α, β > 0 only dependent
on ϵ, δ and γ such that the operator norm satisfies
∥PCn (R(n),{Mj }j ,{Aj }j ) ∥op ≤ αe−βn for n = 0, 1, 2, . . .
|A |
The second theorem shows that bounded quotients such that supj Mjj < 1 is
not itself a sufficient condition for the localization operator to converge to zero in
the operator norm.
Theorem B.4.2. Fix γ > 0, and suppose the radius R satisfies the condition
1/2
πR2 (n) = γ · M1 M2 . . . Mn
.
Then there exist indexed bases {Mj }j and alphabets {Aj }j with
all j ∈ N such that
lim inf ∥PCn (R(n),{Mj }j ,{Aj }j ) ∥op > 0.
n→∞
68
|Aj |
Mj
≤ ϵ < 1 for
B.4. Further generalizations: Indexed Cantor set
B.4.2
Set-up and simple example
Based on our analysis of its simpler sister-operator PCn (R,M,A ) , we begin by deducing some analogous results for PCn (R,{Mj }j ,{Aj }j ) . In particular, identity (B.3.2)
can be replaced by
λ0 (Cn+1 (R, {Mj }j , {Aj }j )) = A0,Mn+1 ,An+1 (·, πR2 (M1 M2 . . . Mn )−1 )
(B.4.4)
·λ0 (Cn (R, {Mj }j , {Aj }j )).
Further, as a matter of additional indexing in the proofs, it follows that an analogue
of Theorem B.3.1 must hold for the indexed localization operator, namely
∥PCn (R,{Mj }j ,{Aj }j ) ∥op ≤ 2λ0 (Cn (R, {Mj }j , {Aj }j )) for n = 0, 1, 2, . . . ,
By the above inequality, it suffices to establish decay conditions for the first eigenvalue with canonical alphabets. Therefore, by the recursive relation (B.4.4), the
relative areas {A0,Mj ,Aj }j are of particular interest. Recall from result (B.3.4)
that these relative areas only depend on the base Mj and alphabet A j via the
|A |
quotient Mjj . Hence, for simplicity, we set
A |Aj | (T ) := A0,Mj ,Aj (·, T ).
Mj
Recall, by Lemma B.3.4, that the relative area A |Aj | (T ) is a monotonically inM
creasing function for T > 0. This again easily translates to the first eigenvalue
with canonical alphabets being an increasing function as a function of the radius,
i.e.,
λ0 (Cn (R1 , {Mj }j , {Aj }j )) ≤ λ0 (Cn (R2 , {Mj }j , {Aj })) ∀ R1 ≤ R2 .
Thus, in the proof of Theorem B.4.1 (and in Theorem B.4.2), we need only consider
the equality case of condition (B.4.3). From the general formula
2
λ0 (Cn (R(n), . . . )) = 1 − e−πR (n)
·
n
Y
j=1
A |Aj | πR2 (n) · (M1 . . . Mj−1 )−1 ,
(B.4.5)
Mj
we shall consider the eigenvalue on the form
1/2
λ0 (Cn (R(n), . . . )) = 1 − e−γ·(M1 ...Mn )
1/2 !
n
Y
M j . . . Mn
·
A |Aj | γ ·
.
M1 . . . Mj−1
Mj
j=1
(B.4.6)
We conclude this section with a simple example where the operator norm
∥PCn (... ) ∥op does not decay to zero as the iterate n → ∞.
69
Paper B. Daubechies’ localization operator on Cantor type sets II
Example B.4.1. (Cantor set of positive measure) The measure of the n-iterate
indexed Cantor set is given by
|Cn (R, {Mj }j , {Aj }j )| = R
n
Y
|Aj |
.
Mj
j=1
|A |
Hence, if Mjj → 1 as j → ∞ at a sufficient rate, the above product does not
converge to zero as n → ∞, and so the Cantor set itself has positive measure.
That is, for such a choice of bases {Mj }j and alphabets {Aj }j , there exists some
0 < ρ < 1 for which |Cn (R, {Mj }j , {Aj }j )| ≥ R · ρ > 0 for all n = 0, 1, 2, . . . Using
the integral formula for the first eigenvalue as a lower bound for the operator norm,
it follows that
Z
∥PCn (R,{Mj }j ,{Aj }j ) ∥op ≥
e−r dr
Cn (πR2 ,... )
−πR2
≥e
2
Cn (πR2 , . . . ) ≥ e−πR R · ρ ∀ n ∈ N.
This shows that for a Cantor set of positive measure lim inf n ∥PCn (R,... ) ∥op > 0
when the radius R is fixed.
If we instead let R = R(n) increase according to the iterates n, we can
choose canonical alphabets Aj = Aj , which guarantees that the first eigenvalue
λk (R(n), . . . ) is also increasing. Hence, lim inf n ∥PCn (R(n),Mj ,Aj ) ∥op > 0.
B.4.3
Proofs of Theorem B.4.1 and Theorem B.4.2
Since the first factor in (B.4.6) tends to 1 as nQ→ ∞, convergence of the operator
n
norm is determined by the remaining product j=1 (. . . ). For simpler notation we
assume the constant γ = 1 in both proofs.
Proof. (Theorem B.4.1) Initially, note that by the interpretation as a relative area,
the function A |Aj | must also be monotonically increasing in terms of the index
Mj
|Aj |
Mj .
Since the index is bounded by ϵ, we obtain the basic inequality
n
Y
j=1
A |Aj |
Mj
M j . . . Mn
M1 . . . Mj−1
1/2 !
≤
n
Y
j=1
Aϵ
M j . . . Mn
M1 . . . Mj−1
1/2 !
.
From the second condition Mj ∈ [M, M 1+δ ], we determine an upper bound for
each argument in the relative area Aϵ (. . . ). More precisely, we maximize the
factors in the numerator, Mj , . . . , Mn ≤ M 1+δ , and minimize the factors in the
denominator, M1 , . . . , Mj−1 ≥ M , so that
Mj . . . Mn
≤ M (1+δ)(n−(j−1)) · M −(j−1)
M1 . . . Mj−1
= M (1+δ)n−(2+δ)(j−1) for j = 1, 2, . . . , n.
70
B.4. Further generalizations: Indexed Cantor set
By monotonicity of Aϵ , it follows that
1/2 ! n−1
n
Y
Y
M j . . . Mn
≤
A |Aj |
Aϵ M [(1+δ)n−(2+δ)j]/2 .
M1 . . . Mj−1
Mj
j=1
j=0
We now define the set of factors which yields negative exponents in the argument
M [... ]/2 , i.e.,
Sn := j ∈ {0, 1, . . . , n − 1} (1 + δ)n − (2 + δ)j ≤ 0 .
(B.4.7)
For these factors the argument satisfies M [... ]/2 ≤ 1, which, by monotonicity,
means
Y
Y
Aϵ M [... ]/2 ≤
Aϵ (1) = Aϵ (1)|Sn | ,
j∈Sn
j∈Sn
where |Sn | denotes the cardinality of Sn . Since the relative areas Aϵ (T ) ∈ (0, 1)
for all T > 0, the remaining factors in {0, 1, . . . , n − 1} \ Sn can be bounded by
1. Hence, in order to prove exponential decay in the operator norm, it suffices to
verify that |Sn | ≥ a · n for some constant a > 0. The latest claim is easily verified
by a closer inspection of definition (B.4.7), where Sn can be expressed as
1+δ
1+δ
Sn =
n ,
n + 1, . . . , n − 1 .
2+δ
2+δ
1+δ
Thus, we conclude that |Sn | ∼ 1 − 2+δ
n.
Proof. (Theorem B.4.2) We will construct a sequence of bases {Mj }j and alphabet
cardinalities {|Aj |}j where the product in (B.4.6) does not converge to zero as the
iterate n tends to infinity:
For any fixed ϵ ∈ (0, 1), chose an initial base M > 1 and alphabet size |A | < M
|
such that |A
M ≤ ϵ. Set M1 := M and |A1 | := |A |, and for j > 1 set
Mj := M1 M2 . . . Mj−1 and |Aj | :=
|A |
Mj ∈ N
M
|A |
so that the fraction Mjj remains constant and equal to
construction, the product in (B.4.6) reads
1/2 !
n
Y
M j . . . Mn
Aθ
M1 . . . Mj−1
j=1
|A |
M
=: θ. With this
n
Y
= Aθ [M1 . . . Mn ]1/2
Aθ [Mj+1 . . . Mn ]1/2
j=2
(B.4.8)
= Aθ [M1 . . . Mn ]1/2
· Aθ [M3 . . . Mn ]1/2 . . . Aθ [Mn−1 Mn ]1/2 Aθ Mn1/2 Aθ (1) .
Proceeding, we make the following two basic observations
71
Paper B. Daubechies’ localization operator on Cantor type sets II
(i) Aθ (T ) ≥ 1 − e−θT (which is still monotonically increasing) and
(ii) Mj ≥ M ∀ j ∈ N.
Combining these two observations with product (B.4.8), then yields the lower
bound
n
Y
n ∞ Y
Y
m
m
Aθ [Mj+1 . . . Mn ]1/2 ≥
1 − e−θN
≥
1 − e−θN
,
m=2
j=2
m=2
where we have defined N := M 1/2 for simplicity. It remains to show that the righthand-side of the above inequality does not converge to zero for any fixed N > 1
and θ ∈ (0, 1). Exchanging the product for a sum, the statement is equivalent to
showing that
∞
X
m
ln 1 − e−θN
> −∞ for any fixed N > 1 and θ ∈ (0, 1).
m=2
Utilizing the lower bound ln(1 − x) ≥ −
∞
X
x
1−x
for 0 < x < 1, we obtain
∞
X
m
ln 1 − e−θN
≥−
m=2
m
∞
X
m
e−θN
−θ −1
≥
−
1
−
e
e−θN ,
m
−θN
1−e
m=2
m=2
where the last sum converges by comparison with the geometric series.
Acknowledgements
The research of the author was supported by Grant 275113 of the Research Council
of Norway. The author would like to extend thanks to Eugenia Malinnikova for
many insightful discussions and feedback on early drafts of the manuscript.
72
Appendix
B.A
Omitted proof in Section B.2.2: Weak
subadditivy
Lemma B.A.1. Let Gn,M,A denote the Cantor-function, defined in (B.2.8). Then
for any x ≤ y,
Gn,M,A (y) − Gn,M,A (x) ≤ Gn,M,A (y − x).
(B.A.1)
Proof. Initially, note that the quantity Gn,M,A (y) − Gn,M,A (x) measures the portion of the n-iterate Cantor set Cn contained in the interval I = [x, y]. Since the
n-iterate is based in the interval [0, 1], we only need to consider the case when
I ⊊ [0, 1].
By definition, we have that the n-iterate Cantor set Cn effectively partitions
n
[0, 1] into M n subintervals {Ikn }M
k=1 , which we will refer to as the n-blocks. These
blocks will either belong to Cn or have zero intersection, i.e.,
|Ikn ∩ Cn | = |Ikn |(= M −n ) or = 0.
By the recursive construction, it follows that Ikn−1 ∩ Cn either contains an entire
alphabet of n-blocks or no n-blocks at all, i.e.,
|Ikn−1 ∩ Cn | = |A |M −n or = 0.
In general,
(i) an (n − j)-block Ikn−j satisfies |Ikn−j ∩ Cn | = |A |j M −n or = 0.
Based on the distribution of the canonical alphabet, we deduce that
(ii) for any positive integer m and positive parameter a,
n
o
Gn,M,A (mM j−n + a) = min 1, Gn,M,A (mM j−n ) + Gn,M,A (a) .
Furthermore, utilizing result (i) and normalizing the Cantor function,
(iii) for m ≤ M consecutive (n − j)-blocks,
Gn,M,A (mM j−n ) = min{m, |A |}|A |j−n .
Since the measure of any interval I ⊊ [0, 1] can be uniquely expressed as
|I| = a +
n−1
X
j=1
73
mj M j−n
(B.A.2)
Paper B. Daubechies’ localization operator on Cantor type sets II
for integers 0 ≤ mj < M and parameter 0 ≤ a ≤ M 1−n , we are now able to
derive a more explicit formula for Gn,M,A . In particular, by recursive use of (ii)
and finally inserting result (iii), we obtain
Gn,M,A (|I|) = Gn,M,A a +
n−1
X
mj M j−n
j=1


n−1


X
Gn,M,A (mj M j−n
= min 1, Gn,M,A a) +


j=1


n−1


X
= min 1, min{aM n , |A |}|A |−n +
min{mj , |A |}|A |j−n .


(B.A.3)
j=1
Thus, it remains to show that the left-hand side of inequality (B.A.1) is bounded
by (B.A.3). For this purpose, we shall utilize the following results:
(iv) An interval of size |I| ≤ M 1−n satisfies
Gn,M,A (y) − Gn,M,A (x) ≤ min{|I|M n , |A |}|A |−n .
(v) An interval of size |I| ≤ M j−n for j ∈ N contains at most |A |j n-blocks, i.e.,
Gn,M,A (y) − Gn,M,A (x) ≤ |A |j−n .
In the case (iv), observe that I will at most intersect two (n − 1)-blocks, say Ik and
Ik+1 . If both intersections of Ik and Ik+1 with Cn is non-zero, then the alphabet
of n-blocks will have the same distribution in each (n − 1)-block. Hence, by (i),
|I ∩ Cn | = |I ∩ Ik ∩ Cn | + |(I − M 1−n ) ∩ Ik ∩ Cn | ≤ |Ik ∩ Cn | = |A |M −n ,
which yields (iv) after normalization. Result (v) follows by a similar argument.
Based on observation (v), consider the case when (m−1)M j−n < |I| ≤ mM j−n
for some integer 0 < m ≤ M . Set x0 := x and xm := y and partition the interval
j−n
I = [x, y] = ∪m−1
. Then
l=0 [xl , xl+1 ] such that xl < xl+1 and xl+1 − xl ≤ M
(vi) Gn,M,A (y) − Gn,M,A (x) =
m−1
X
Gn,M,A (xl+1 ) − Gn,M,A (xl )
l=0
≤ min{m, |A |}|A |j−n .
Finally, for an arbitrary interval I = [x, y] ⊆ [0, 1], we utilize the partitioning
associated with (B.A.2) along with results (iv)-(vi) to conclude that
Gn,M,A (y) − Gn,M,A (x)
is indeed bounded from above by (B.A.3).
74
Paper C
A fractal uncertainty principle for the short-time
Fourier transform and Gabor multipliers
Helge Knutsen
Published in
Applied and Computational Harmonic Analysis, 2023, volume 62.
Paper C
A fractal uncertainty principle for
the short-time Fourier transform
and Gabor multipliers
Abstract
We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with
Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire
functions. Specifically for signals in L2 (Rd ), we obtain norm estimates for
Daubechies’ time-frequency localization operator localizing on porous sets.
The proof is based on the maximal Nyquist density of such sets, which we
also use to derive explicit upper bound asymptotes for the multidimensional
Cantor iterates, in particular. Finally, we translate the fractal uncertainty
principle to discrete Gaussian Gabor multipliers.
C.1
Introduction
The fractal uncertainty principle (FUP) was first introduced and developed for the
separate time-frequency representation in [29], [12], [28], see also [61] for explicit
estimates. It states that no signal in L2 (R) can be concentrated near fractal sets in
both time and frequency. We reference Dyatlov’s detailed introduction to the topic
[27], where fractal sets are defined broadly in terms of either δ-regular sets or almost
equivalently in terms of ν-porous sets within the scale bounds 0 < h to 1 (see Def.
C.2.1). In both definitions, Dyatlov considers families of subsets T (h), Ω(h) ⊆ [0, 1]
and formulates the FUP for said families as the lower bound scale h → 0. The
FUP is presented as a norm estimate for the localization operator χΩ(h) Fh χT (h) ,
where χE denotes the characteristic function of a subset E, and Fh denotes the
√ −1
dilated Fourier transform Fh f (ω) := h F f (ωh−1 ). In terms of ν-porosity (see
Theorem 2.16 in [27]), for signals in L2 (R) and for families T (h), Ω(h) ⊆ [0, 1] of
77
Paper C. An FUP for the STFT and Gabor multipliers
ν-porous sets on scales h to 1, there exist constants C, β > 0 only dependent on
ν > 0 such that
∥χΩ(h) Fh χT (h) ∥L2 (R)→L2 (R) ≤ Chβ ∀ 0 < h ≤ 1.
√
Alternatively, if we disentangle h from the Fourier transform and write h as h,
we obtain a statement with regard to families T (h), Ω(h) ⊆ [0, h−1 ] of ν-porous
sets on scales h to h−1 , to which there exist constants C, β > 0 only dependent on
ν > 0 so that
∥χΩ(h) F χT (h) ∥L2 (R)→L2 (R) ≤ Chβ ∀ 0 < h ≤ 1.
(C.1.1)
On this form, the FUP more clearly reads as an uncertainty principle as, depending
on our choice of ν, the measures of our time and frequency set, |T (h)| and |Ω(h)|,
respectively, might tend to infinity as h → 0.
Inspired by the FUP in the separate representation and motivated by the understanding that uncertainty principles should be present regardless of time-frequency
representation, we search for analogous results in the joint representation. In particular, we consider and have considered Daubechies’ localization operator PΩ
based on the Short-Time Fourier Transform (STFT) with the Gaussian window
that projects onto some subset Ω of the time-frequency plane. In previous installments [62], [63], we have restricted our attention to radially symmetric subsets in
R2 , as this yields a known eigenbasis, the Hermite functions, and explicit formulas for the associated eigenvalues. With such insights, we have been able to derive
estimates for the operator norm when localizing on radial Cantor iterates that mirror estimate (C.1.1) but with explicit estimates for the exponent, sometimes even
precise estimates. The radial assumption has also proved effective for Bergman
spaces and by extension for analytic wavelets in [5], where direct knowledge of
the eigenvalues of the localization operator have produced similar estimates when
localizing on the mid-third radial Cantor set.
In the present paper, however, we abandon the radial assumption and instead
consider the more general problem of optimal localization on ν-porous sets in
phase space R2d for arbitrary d ≥ 1. Although we no longer have knowledge of
the eigenvalues of such an operator, the Gaussian window in the STFT in and of
itself provides additional structure. Namely, Daubechies’ operator PΩ : L2 (Rd ) →
L2 (Rd ) can equivalently be viewed as a Toeplitz operator on Bargmann-Fock spaces
or simply Fock spaces, F 2 (Cd ), of square integrable entire functions with respect to
a Gaussian measure. With this perspective, we utilize the subaveraging properties
of entire functions to derive estimates of the operator norm ∥PΩ ∥op in terms of the
maximal Nyquist density of Ω. These estimates bear resemblance to the estimates
in Abreu and Speckbacher’ paper [7], which in large part served as inspiration for
our approach. By an inductive scheme, we find that for a family Ω(h) ⊆ R2d of
ν-porous sets on scales h to 1, there exists constants C, β > 0 only dependent on
ν > 0 (and d) such that
∥PΩ(h) ∥op ≤ Chβ ∀ 0 < h ≤ 1,
(C.1.2)
which represents a direct analogue of (C.1.1), now in the joint representation.
In fact, these estimates extend to norm estimates on the Fock space F p (Cd ) for
78
C.2. Preliminaries
generic p ≥ 1, which in turn yield an FUP not only for L2 (Rd ) but also for the modulation spaces, M p (Rd ). For more explicit estimates of the exponent in (C.1.2), we
specifically consider multidimensional Cantor set constructions, and here the upper bound asymptotes rely on our ability to directly compute the Nyquist density
of such sets.
In addition, we present an FUP for Gabor multipliers, which represent a discrete alternative to Daubechies’ localization operator based on Gabor frames (see
[36] for an introduction to Gabor multipliers). Approximation properties of such
operators have been studied in [48], [19], and spectral properties have been studied
[37]. For our purpose, we consider the closest comparison to the Daubechies’ operator with a Gaussian window. Namely, we consider the case when the generating
function of the Gabor multiplier also equals a Gaussian.
The paper is organized as follows: Section C.2 contains necessary background
theory. This includes, a formal introduction to the modulation spaces, Fock spaces
and their connection to the STFT (Section C.2.1 and C.2.2), an introduction to
Gabor frames and Gabor multipliers (Section C.2.3) and a precise description of
what we mean by ”fractal” with the Cantor set constructions as concrete examples
(Section C.2.4). The results are divided into three sections C.3, C.4 and C.5. The
general FUP for Fock spaces and modulation spaces are formulated in Section C.3.
The next Section C.4 is focused on the multidimensional Cantor set constructions,
with an FUP formulated specifically for these sets. In the last Section C.5 we show
how the FUP can be translated to Gaussian Gabor multipliers.
C.2
C.2.1
Preliminaries
The short-time Fourier transform and modulation
spaces
Consider some fixed window function ϕ : Rd → C, to which we introduce the basic
operators Tx ϕ(t) := ϕ(t − x) and Mω ϕ(t) := e2πiω·t ϕ(t), i.e., time-translation and
frequency-modulation, respectively. The Short-Time Fourier Transform (STFT)
of some signal f ∈ L2 (Rd ), with respect to window ϕ, evaluated at a point (x, ω) ∈
Rd × Rd , is then given by the inner product
Vϕ f (x, ω) := ⟨f, Mω Tx ϕ⟩.
Observe that if ϕ ≡ 1, the STFT coincides with the regular Fourier transform.
For non-constant windows, however, we obtain a joint time-frequency description
of our signal. Furthermore, for ∥ϕ∥2 = 1, the STFT becomes an isometry onto
some subspace of L2 (R2d ), i.e., ⟨Vϕ f, Vϕ g⟩L2 (R2d ) = ⟨f, g⟩. In this case, we have an
inversion formula, namely
Z
f=
Vϕ f (x, ω)Mω Tx ϕ dxdω,
(C.2.1)
R2d
where the integral is interpreted in the weak-sense. Daubechies’ time-frequency
localization operator, PSϕ : L2 (Rd ) → L2 (Rd ), with some bounded symbol S, is then
79
Paper C. An FUP for the STFT and Gabor multipliers
obtained by modifying the above integrand by the multiplicative weight function
S(x, ω), i.e.,
Z
ϕ
PS f :=
S(x, ω) · Vϕ f (x, ω)Mω Tx ϕ dxdω
R2d
⇐⇒
⟨PSϕ f, g⟩
= ⟨S · Vϕ f, Vϕ g⟩L2 (R2d ) ∀ g ∈ L2 (Rd ).
(C.2.2)
This could equivalently be viewed as modifying the resulting STFT by multiplication by S before inversion. Oftentimes, we consider S = χΩ , i.e., the characteristic
function of a subset Ω of the phase space R2d , so that the operator PΩϕ := PχϕΩ is
interpreted as projecting signals onto said time-frequency domain. The associated
operator norm ∥PΩϕ ∥op then measures the optimal localization on Ω.
For general p ≥ 1, we do not consider Lp -signals, rather, we consider signals in
the modulations spaces, introduced in [35]. For this purpose, we fix the window
ϕ ̸≡ 0 in the Schwartz class S (Rd ), and we denote the associated dual space of
tempered distributions by S ′ (Rd ). The modulation space M p (Rd ) is then defined
as the subspace
M p (Rd ) := f ∈ S ′ (Rd ) ∥Vϕ f ∥Lp (R2d ) < ∞ ⊆ S ′ (Rd ).
Depending on our particular choice of window, we induce equivalent norms on
M p (Rd ), namely ∥f ∥M p := ∥Vϕ f ∥Lp (R2d ) . Again, we may consider localization on
some subset Ω ⊆ R2d , using the modulation spaces. More precisely for any fixed
p ≥ 1, we consider the quantity
sup
f ∈M p (Rd )\{0}
∥Vϕ f · χΩ ∥pLp (R2d )
∥Vϕ f ∥pLp (R2d )
(C.2.3)
as a measure of optimal localization on Ω. Notably for p = 2, we have that
M 2 (Rd ) = L2 (Rd ), and we retrieve the same localization estimate using Daubechies’
localization operator, where ∥PΩϕ ∥op coincides with (C.2.3).
C.2.2
From modulation spaces to Fock spaces
A popular choice for window function is the Gaussian function, which on Rd reads
2
ϕ0 (x) := 2d/4 e−πx , where x2 = x21 + · · · + x2d for x = (x1 , . . . , xd ).
With this window choice, we can, in fact, rephrase the localization estimate (C.2.3)
for the modulation spaces M p (Rd ) as estimates in the Bargmann-Fock space or
simply Fock space, F p (Cd ). In particular, Daubechies’ localization operator can
be replaced by a Toeplitz operator on F 2 (Cd ).
We reference Zhu’s book [87] for a detailed introduction to Fock spaces in C.
For a complex vector z = x + iω = (z1 , . . . , zd ) ∈ Cd , we distinguish between
z 2 = z · z = z12 + · · · + zd2 and |z|2 = z · z = |z1 |2 + · · · + |zd |2 . Now, for arbitrary
p
2
p ≥ 1, let dµp (z) := e− 2 π|z| dA(z) denote the Gaussian measure on Cd , where
dA(z) is the volume measure dxdω. The associated Lp -space, Lp (Cd , dµp ), is
80
C.2. Preliminaries
simply denoted by Lp (Cd ). The Fock space F p (Cd ) is then defined as the Banach
space of entire functions F ∈ Lp (Cd ), with norm1
Z
1/p
p
2
∥F ∥Lp =
|F (z)|p e− 2 π|z| dA(z)
< ∞.
Cd
For p = ∞, we let L∞ (Cd ) denote the space of measurable functions on Cd such
that
o
n
2
π
∥F ∥L∞ = ess sup |F (z)|e− 2 |z| z ∈ Cd < ∞.
Again, the Fock space F ∞ (Cd ) is the Banach space of entire functions in L∞ (Cd ).
For p = 2, we find that the Fock space forms a reproducing kernel Hilbert space,
with reproducing kernel Kξ (z) = eπz·ξ so that ⟨F, Kξ ⟩L2 = F (ξ). Utilizing this
kernel, we obtain an orthogonal projection P : L2 (Cd ) → F 2 (Cd ), defined by
Z
2
P F (z) :=
Kξ (z)F (ξ)e−π|ξ| dA(ξ).
Cd
For a bounded measurable function S : Cd (∼
= R2d ) → C, we define the Toeplitz
operator TS : F 2 (Cd ) → F 2 (Cd ), with symbol S, by
Z
2
TS F (z) := P (SF )(z) =
S(ξ) · Kξ (z)F (ξ)e−π|ξ| dA(ξ).
Cd
If we initially consider test functions G ∈ L1 (Cd ) ∩ F 2 (Cd ), then, by Fubini’s
theorem and the reproducing property of the kernel Kξ (z), the inner product
attains the simple form
⟨TS F, G⟩L2 = ⟨S · F, G⟩L2
(C.2.4)
By a density argument, it follows that (C.2.4) holds for all G ∈ F 2 (Cd ), more akin
to the inner product (C.2.2).
The precise connection to the modulation spaces and Daubechies’ localization
operator with a Gaussian window is established through the Bargmann transform.
The Bargmann transform was first introduced in [8] as an isometric isomorphism
B : L2 (Rd ) → F 2 (Cd ), given by
Z
2
π 2
B f (z) := 2d/4
f (t)e2πt·z−πt − 2 z dt.
Rd
Later, in [78] and [2], more general mapping properties of the Bargmann transform
have been investigated, where, in fact, the transform has been shown to extend
to an isometric isomorphism B : M p (Rd ) → F p (Cd ). The Bargmann transform is
closely related to the STFT with a Gaussian window Vϕ0 , where it is straightforward to verify the following identity
π
2
Vϕ0 f (x, −ω) = eπix·ω B f (z)e− 2 |z|
for z = x + iω.
(C.2.5)
1 For consistent and simple notation, we denote the norm in the Fock space by ∥ · ∥ p rather
L
than ∥ · ∥F p . In particular, this is to avoid switching notation for functions in Lp (Cd ) \ F p (Cd ),
e.g., when we consider F · χΩ for F ∈ F p (Cd ) \ {0} and Ω ⊊ Cd .
81
Paper C. An FUP for the STFT and Gabor multipliers
Utilizing the above identity, we define S ∗ (z) := S(z) so that the inner product ⟨PSϕ0 f, g⟩ = ⟨TS ∗ B f, B g⟩L2 , and the one-to-one correspondence between the
Daubechies localization operator PSϕ0 and the Toeplitz operator TS ∗ becomes apparent. In particular, their norms coincide. Moreover, for arbitrary p ≥ 1, we find
that
∥Vϕ0 f · χΩ ∥Lp (R2d ) = ∥B f · χΩ∗ ∥Lp ,
(C.2.6)
where Ω∗ := {z ∈ Cd | z ∈ Ω} denotes the complex conjugate subset.
In the subsequent discussion, we shall therefore consider norm estimates in the
Fock spaces. For p = 2, we consider Toeplitz operators projecting onto Ω ⊆ Cd ,
which we will simply denote by TΩ = TχΩ . By (C.2.4), the associated operator
norm is given by
Z
2
∥TΩ ∥op = sup
|F (z)|2 e−π|z| dA(z).
∥F ∥F 2 =1
Ω
For general p ≥ 1, we consider upper bounds for the quantity
R
p
2
|F (z)|p e− 2 π|z| dA(z)
∥F · χΩ ∥pLp
Ω
=
with F ∈ F p (Cd ) \ {0}.
∥F ∥pLp
∥F ∥pLp
By the use of complex interpolation (see Appendix C.A for details), the above
quotients are actually bounded by the estimate for p = 1, i.e.,
∥F · χΩ ∥pLp
∥F · χΩ ∥L1
≤
sup
,
p
∥F
∥
∥F ∥L1
p
d
1
d
p
F ∈F (C )\{0}
F ∈F (C )\{0}
L
sup
(C.2.7)
which allows for estimates without any p-dependence. Unsurprisingly, the added
structure provided by the Fock space turns out to be beneficial when estimating the
norm. Namely, we shall exploit subaveraging properties of subharmonic functions.
C.2.3
Gabor frames and Gabor multipliers
In general, a family of vectors {ϕλ }λ∈Λ in the Hilbert space H is called a frame if
there exist constants 0 < A ≤ B < ∞, i.e., frame bounds, such that
X
A∥f ∥2 ≤
|⟨f, ϕλ ⟩|2 ≤ B∥f ∥2 ∀ f ∈ H.
λ∈Λ
P
The associated frame operator S : H → H is given by S f := λ ⟨f, ϕλ ⟩ϕλ with
norm A ≤ ∥S ∥H ≤ B, and ∥S ∥H = B if B is the optimal upper frame bound.
If A = B, the √
frame is called a tight frame. By renormalizing the vectors so
that ϕλ 7→ ϕλ / A, any tight frame can be turned into
P a Parseval frame, i.e.,
A = B = 1, where we also have the representation f = λ ⟨f, ϕλ ⟩ϕλ , i.e., S = id.
The Gabor frame for L2 (Rd ) is based on the idea of discretizing the STFT
inversion formula (C.2.1). For this purpose, consider a lattice Λ ⊆ R2d of sampling
points. Oftentimes, we consider rectangular lattices of the form Λ = aZd ×bZd with
82
C.2. Preliminaries
parameters a, b > 0. Further, fix a window function ϕ ∈ L2 (Rd )\{0} and define the
time-frequency shifts π(x, ω)ϕ := Mω Tx ϕ. If the family of time-frequency shifts
{π(λ)ϕ}λ∈Λ forms a frame, we call this system a Gabor frame with generating
function ϕ over the lattice Λ. We may also include a normalization factor based
on the density of the lattice. More precisely, to each lattice we can associate a
connected neighbourhood of the origin DΛ called the fundamental region such that
∪λ∈Λ (λ + DΛ ) = R2d and |(λ + DΛ ) ∩ (ξ + DΛ )| = 0 whenever λ ̸= ξ ∈ Λ.
With the normalization
L2 (Rd ) reads
p
|DΛ |{π(λ)ϕ}λ∈Λ , the frame operator SΛϕ : L2 (Rd ) →
SΛϕ f = |DΛ |
X
⟨f, π(λ)ϕ⟩π(λ)ϕ.
(C.2.8)
λ∈Λ
We immediately recognize SΛϕ f p
as a Riemann sum of the integral (C.2.1). Thus,
for a sequence of Gabor frames |DΛn |{π(λ)ϕ}λ∈Λn where |DΛn | → 0 as n → ∞,
we expect the discretization (C.2.8) to converge weakly to the integral (C.2.1)
and the frame bounds
to tighten. Notably for the sequence of square lattices
Λn = n1 Zd × Zd and f, ϕ in Feichtinger’s algebra M 1 (Rd ), Weisz shows in [84]
(Theorem 2) that SΛϕn f converges to f in the M 1 -norm as n → ∞. In the same
paper, while ϕ remains in M 1 (Rd ), the result is extended (Theorem 5) to f in the
more general modulation space M p,q (Rd ) with convergence in the M p,q -norm. In
particular, since M 2 (Rd ) = L2 (Rd ), we have convergence in the L2 -norm when
f ∈ L2 (Rd ).
ϕ
The Gabor multiplier, GΛ,b
: L2 (Rd ) → L2 (Rd ), represents a discretization of
Daubechies’ operator in (C.2.2), where the sum (C.2.8) is weighted by a bounded
symbol b defined on the lattice Λ, i.e.,
X
ϕ
GΛ,b
f := |DΛ |
b(λ)⟨f, π(λ)ϕ⟩π(λ)ϕ.
λ∈Λ
For localization on a specific subset Ω ⊆ R2d , we shall consider Gabor symbols b
that mimic the behaviour of the characteristic function χΩ . One natural option is
to consider the portion of a lattice point region λ + DΛ containing the subset Ω,
i.e.,
bΩ (λ) :=
|Ω ∩ (λ + DΛ )|
∈ [0, 1].
|DΛ |
Alternatively, we only distinguish between whether the lattice point region λ + DΛ
contains a non-zero part of Ω. That is, we apply the ceiling function ⌈·⌉, rounding
up to the nearest integer, so that
⌈bΩ (λ)⌉ ∈ {0, 1}.
Evidently 0 ≤ bΩ (λ) ≤ ⌈bΩ (λ)⌉, from which it is easily verified that the operϕ
ϕ
ator norms also satisfy ∥GΛ,b
∥op ≤ ∥GΛ,⌈b
∥op . Thus, when estimating upper
Ω
Ω⌉
83
Paper C. An FUP for the STFT and Gabor multipliers
bounds for the operator norm in Section C.5, we only consider the second symbol
suggestion, ⌈bΩ ⌉. Notice that utilizing symbol ⌈bΩ ⌉ is the same as restricting the
summation (C.2.8) to a subset of the lattice Λ, namely
(C.2.9)
ΛΩ := λ ∈ Λ |Ω ∩ (λ + DΛ )| > 0 .
ϕ
For simplicity, we denote the Gabor multiplier with symbol ⌈bΩ ⌉ by GΛ,Ω
, which,
by the above observation, is given by
X
ϕ
⟨f, π(λ)ϕ⟩π(λ)ϕ.
(C.2.10)
GΛ,Ω
f = |DΛ |
λ∈ΛΩ
C.2.4
Porous sets and Cantor sets
We shall define ”fractal sets” in terms of the general notion of ν-porosity. It is
based on Def. 2.7 in [27], adjusted to higher dimensions. Informally, in order for
a set to be classified as porous, we require the set to contain gaps or pores within
certain scale bounds.
Def. C.2.1. (ν-porosity) Suppose 0 < ν < 1, 0 ≤ αmin ≤ αmax ≤ ∞ and Ω ⊆ Rd
is closed. We say that Ω is ν-porous on scales αmin to αmax if for every ball Br (x)
of radius r ∈ [αmin , αmax ] there exists a ball Bνr (y) ⊆ Br (x) of radius νr such that
|Ω ∩ Bνr (y)| = 0.
Notice that in one dimension, the ν-porous set resembles a Cantor type set,
where we are able to remove a ν-portion of any interval inductively down to the
lower bound scale. The Cantor sets represent a popular and easy to understand
family of fractal sets, which we construct as follows:
Let M > 1 be a fixed integer, and let A be a non-empty proper subset of
{0, 1, . . . , M − 1}. The n-iterate (n-order) discrete Cantor set with base M and
alphabet A is then defined as


n−1

X
Cn(d) (M, A ) :=
aj M j aj ∈ A , j = 0, 1, . . . , n − 1 ⊆ {0, 1, . . . , M n − 1}.


j=0
The ”continuous” n-iterate Cantor set based in the interval [0, L] is given by
Cn (L, M, A ) := LM −n · Cn(d) (M, A ) + [0, LM −n ] for n = 0, 1, 2, . . .
The iterates are nested, i.e., Cn+1 ⊆ Cn , and the (limit) Cantor set is then given
by the intersection of all the n-iterates. While the Cantor set itself has measure
zero, each n-iterate does not. If we let |A | denote the cardinality of the alphabet
A , the measure of the n-iterate Cantor set is given by
n
|A |
|Cn (L, M, A )| =
L.
M
84
C.2. Preliminaries
Note that for M = 3 and A = {0, 2}, we obtain the standard mid-third n-iterate
Cantor set, with measure (2/3)n L.
Unsurprisingly, the Cantor sets are indeed ν-porous. Below we present a simple
estimate for the porosity constant and the scales (see Appendix C.B for details):
Lemma C.2.1. The n-iterate Cantor set Cn (L, M, A ) with base M > 1 and
alphabet size |A | < M , based in the interval [0, L], is ν-porous on scales LM −n+1
to ∞, with any ν ≤ 12 M −2 .
In multiple dimensions R2d , we consider two possible Cantor set constructions:
1. For a ball of radius R > 0 centered at the origin, we consider the radially
symmetric n-iterate Cantor set as a subset of the form
o
n
d
x21 + · · · + x22d ∈ Cn (R2d , M, A ) .
Cn2d (R, M, A ) := (x1 , . . . , x2d ) ∈ R2d
In particular, localizing on Cn2 (R, M, A ) has been discussed extensively in
[63]. In general, these radially symmetric Cantor iterates are constructed
such that all annuli that make up the set have the same measure. The total
measure is given by
|Cn2d (R, M, A )| = Cn (πR2 )d /d!, M, A
where we recognize
R.
(πR2 )d
d!
=
|A |
M
n
(πR2 )d
,
d!
as the volume of the 2d-dimensional ball of radius
2. Alternatively, we can consider the Cartesian product of 1-dimensional Cantor
iterates
Cn1 (L1 , M1 , A1 ) × Cn2 (L2 , M2 , A2 ) × · · · × Cn2d (L2d , M2d , A2d ).
If all iterates coincide, the above Cartesian product reduces to Cn (L, M, A )2d
based in the hypercube [0, L]2d .
Remark. In [27] fractal sets are originally defined in terms of δ-regularity for
some 0 < δ < 1 (see Def. 2.2 in [27]). While only formulated in 1-dimension, the
notion of δ-regularity can also be extended to higher dimensions. Compared to
ν-porosity, this concept offers a different perspective on fractal sets: For instance
with regard to the n-iterate Cantor set Cn (L, M, A ), we find that the δ corre|
sponds to fractal dimension (or Hausdorff dimension) of the iterate, namely lnln|A
M .
However, the notion of δ-regularity might appear more abstract than ν-porosity as
it does not immediately read as a set containing gaps, and less so the size of those
gaps. Nonetheless, as shown in [27] Proposition 2.10, any δ-regular set is ν-porous,
and the scales associated to δ-regularity coincide with the ν-porous scales up to
multiplicative constants. Thus, formulating the FUP on ν-porous sets directly
translates to an FUP on δ-regular sets.
85
Paper C. An FUP for the STFT and Gabor multipliers
C.3
Fractal uncertainty principle in joint
representation
In this section we present the FUP for the joint time-frequency representation.
Initially, in Section C.3.1 we derive the FUP for the Fock spaces. In the subsequent
Section C.3.2 we translate the FUP for the Fock spaces to an uncertainty principle
for the STFT on modulation spaces.
C.3.1
Fractal uncertainty principle for Fock spaces
Theorem C.3.1. (FUP for Fock Spaces F p (Cd )) Let 0 < h ≤ 1, and suppose
Ω(h) ⊆ Cd is an h-dependent family of sets which is ν-porous on scales h to 1.
Then for all p ≥ 1 and all F ∈ F p (Cd ) \ {0} there exist constants C, β > 0 only
dependent on ν (and d) such that
∥F · χΩ(h) ∥pLp
≤ Chβ ∀ 0 < h ≤ 1.
∥F ∥pLp
(C.3.1)
In particular, for p = 2, the Toeplitz operator TΩ(h) satisfies
∥TΩ(h) ∥op ≤ Chβ ∀ 0 < h ≤ 1.
(C.3.2)
The essential property of the Fock space that we utilize is the subaveraging
property, where point evaluations can be bounded by an average over the ball up
to a multiplicative constant. The statement can be found in [87] for F p (C), which
generalized to F p (Cd ) (see Appendix C.C for details) reads:
Lemma C.3.2. For any F ∈ F p (Cd ) and any point z ∈ Cd , we have for all R > 0
that

−1
d−1 p
p d
2 j
πR
p
p
2
2 X
2
1 − e− 2 πR

|F (z)|p e− 2 π|z| ≤
2
j!
j=0
(C.3.3)
Z
p
2
·
|F (ξ)|p e− 2 π|ξ| dA(ξ).
BR (z)
Remark. The same statement also appears in [70] and [50], related to the sampling problem in more general Fock spaces, although without an explicit multiplicative constant.
Proceeding, we require the following two concepts: For a set Ω ⊆ Cd and R > 0,
we define the maximal Nyquist density, ρ(Ω, R), by
ρ(Ω, R) := sup |Ω ∩ BR (z)| ≤ max{|Ω|, |BR (0)|}.
z∈Cd
Further, we define the R-thickened set, ΩR , by
ΩR := Ω + BR (0) = ∪z∈Ω BR (z).
86
(C.3.4)
C.3. Fractal uncertainty principle in joint representation
With these notions and Lemma C.3.2, we present an upper bound estimate for the
integral over Ω.
Lemma C.3.3. Suppose Ω ⊆ Cd is measurable. Then for any F ∈ F p (Cd ) and
any R > 0, we have that
Z
p d
p
2
ρ(Ω, R)
|F (z)|p e− 2 π|z| dA(z) ≤
p
2 j
p
2
2 Pd−1 ( 2 πR )
Ω
1 − e− 2 πR
j=0
j!
(C.3.5)
Z
2
π|z|
p −p
·
dA(z).
|F (z)| e 2
ΩR
Proof. By Lemma C.3.2,

Z
|F (z)|p e
2
−p
2 π|z|
dA(z) ≤
p d
2
Ω
1 − e
2
−p
2 πR
p
Since z ∈ Ω, we have that
Z
Z
p
2
|F (ξ)|p e− 2 π|ξ| dA(ξ) =
2
|F (ξ)|p e− 2 π|ξ| dA(ξ)dA(z).
·
Ω
−1

j!
j=0
Z Z
BR (z)
p
2 j
2 πR
d−1
X
BR (z)
p
2
χBR (z) (ξ) · |F (ξ)|p e− 2 π|ξ| dA(ξ).
ΩR
By Fubini’s theorem, we obtain
Z Z
Ω
p
2
|F (ξ)|p e− 2 π|ξ| dA(ξ)dA(z)
BR (z)
Z
Z
=
ΩR
Z
p
2
χBR (ξ) (z)dA(z) |F (ξ)|p e−π 2 |ξ| dA(ξ)
Ω
p
2
|Ω ∩ BR (ξ)| · |F (ξ)|p e− 2 π|ξ| dA(ξ) ≤ ρ(Ω, R)
=
Z
p
2
|F (ξ)|p e− 2 π|ξ| dA(ξ).
ΩR
ΩR
In the above lemma notice that the integral over ΩR is always bounded by the
integral over Cd , that is,
∥F · χΩ ∥pLp ≤
p d
2
1−e
ρ(Ω, R)
∥F ∥pLp ∀ R > 0.
Pd−1 ( p2 πR2 )j
2
−p
2 πR
j=0
j!
Further, by the interpolation result (C.2.7), we can optimize the right-hand side
with the estimate for p = 1 so that
∥F · χΩ ∥pLp ≤ 2−d
1−e
ρ(Ω, R)
∥F ∥pLp ∀ R > 0.
Pd−1 ( 12 πR2 )j
− 12 πR2
j=0
(C.3.6)
j!
87
Paper C. An FUP for the STFT and Gabor multipliers
Regardless, this shows that estimates of the quotient ∥F · χΩ ∥pLp /∥F ∥pLp can be
solely based on estimates of the maximal Nyquist density. E.g., as shown in the
subsequent Section C.4, under certain growth conditions, we are able to obtain
good estimates for the Nyquist density for the standard Cartesian product of
Cantor sets and radial Cantor sets. For the general case of porous sets, however,
we also take into account the integral over the thickened set.
Besides the trivial upper bounds of (C.3.4), if we suppose Ω ⊆ Cd is ν-porous
on scales αmin to αmax and consider a radius R within the scale bounds, we obtain,
by Def. C.2.1, the simple estimate
d
πR2
2d
2d
|BR (0)| = 1 − ν
BR (z) \ BνR (ξ) = 1 − ν
ρ(Ω, R) ≤
sup
.
d!
|z−ξ|≤(1−ν)R
We apply this upper bound to Lemma C.3.3, which yields the corollary:
Corollary C.3.1. Suppose Ω ⊆ Cd is ν-porous on scales αmin to αmax . Then for
any radius R ∈ [αmin , αmax ] and any function F ∈ F p (Cd ), we have that
Z
p
p
2
πR2 · 1 − ν 2d
|F (z)|p e− 2 π|z| dA(z) ≤ κd
2
Ω
Z
(C.3.7)
p
2
|F (z)|p e− 2 π|z| dA(z),
·
ΩR
where the function κd is given by

−1
d−1 j
X
xd 
x
 .
1 − e−x
κd (x) :=
d!
j!
j=0
(C.3.8)
The crucial observation, moving forward, is that for some choices of R > 0,
the thickened set ΩR is itself a porous set provided the original set Ω is porous. A
special case of this observation is presented in Proposition 2.11 [27].
Lemma C.3.4. (Thickening of porous set) Suppose Ω ⊆ Cd is ν-porous on scales
αmin to αmax . For any 0 ≤ r < ν · αmax , consider the r-thickened set Ωr :=
Ω + Br (0) = ∪z∈Ω Br (z). Then for any R ∈ [αmin , αmax ] with νr < R, the set Ωr
is ν − Rr -porous on scales R to αmax .
Proof. Consider αmin ≤ R ≤ L ≤ αmax . By ν-porosity of Ω, for any BL (z) there
exists BνL (ξ) ⊆ BL (z) such that |BνL (ξ)∩Ω| = 0. After r-thickening, we maintain
a zero intersection with Ωr if the radius of BνL (ξ) is reduced to νL − r. Since L
is arbitrary and
r
ν−
L ≤ νL − r =⇒ |B(ν− r )L (ξ) ∩ Ωr | ≤ |BνL−r (ξ) ∩ Ωr | = 0,
R
R
the set Ωr satisfies the claimed porosity properties. The conditions on r and R
ensure that the new porosity is positive and that the scale bounds are valid.
We are now ready to prove Theorem C.3.1, where the main idea is to construct
a sequence of thickened sets, to which we may apply Corollary C.3.1 repeatedly.
88
C.3. Fractal uncertainty principle in joint representation
Proof. (Theorem C.3.1) By the interpolation result (C.2.7), it suffices to prove the
statement for the case p = 1.
Recall that for some 0 < h ≤ 1 we presume Ω = Ω(h) is ν-porous on scales h to
1. For some finite sequence of radii {Rj }nj=1 , we consider an associated sequence
of thickened sets {Ω(j) }n−1
j=0 , given by
Ω(0) := Ω and Ω(j) := ΩR1 +R2 +···+Rj for j = 1, 2, . . . , n − 1.
We chose the radii such that the set Ω(j) maintains the porous property, in the
sense that
Ω(j) is νj -porous on scales Rj+1 to 1.
By Lemma C.3.4, the new porosity constants {νj }j are given by
νj = ν −
R1 + · · · + Rj
R1 + · · · + Rj
for j = 0, 1, . . . , n − 1.
, where Rj+1 >
Rj+1
ν
Thus, the sequence of radii must at least grow exponentially. In particular, we
consider radii of the form
j
3
Rj := h
for j = 1, 2, . . . , n,
(C.3.9)
ν
which not only satisfy the above radii-condition but also yield a positive lower
bound for the porosity constants {νj }j , namely
νj = ν −
j ν j+1 X
3
3
k=1
ν
≥
ν
∀ j ∈ N and 0 < ν < 1.
2
In addition, we will consider a fixed threshold radius 0 < r ≤ 1, specified later,
such that Rj ≤ r for all j = 1, 2, . . . , n.
Proceeding, we make repeated use of Corollary C.3.1 to the sequence of thickened sets {Ω(j) }j based on the radii defined in (C.3.9), all with porosity constant
equal to ν2 . This reveals that the integral over Ω is bounded by
Z
2
π
|F (z)|e− 2 |z| dA(z)
Ω


k
π ν 2d Z
Y
2
π
·
κd
Rj2 · 1 −
|F (z)|e− 2 |z| dA(z)
≤
2
2
(k)
Ω
j=1


k
π ν 2d Y
2

 · ∥F ∥L1 for k = 1, 2, . . . , n − 1. (C.3.10)
≤
κd
Rj · 1 −
2
2
j=1
Since limx→0 κd (x) = 1 and the derivative κ′d (x) > 0 for all x > 0, we may fix a
threshold radius r > 0 such that
π ν 2d κd
R2 · 1 −
< 1 ∀ 0 < R ≤ r.
2
2
89
Paper C. An FUP for the STFT and Gabor multipliers
2d
In particular, fix some 0 < ϵ ≤ ν2
and consider the threshold r := min(1, rϵ ),
where rϵ > 0 denotes the (unique) solution to κd π2 rϵ2 = 1 + ϵ. We shall only
consider the radii Rj ≤ r, for which
π ν 2d ν 4d
ν 2d 2
≤ (1 + ϵ) · 1 −
≤1−
.
(C.3.11)
Rj · 1 −
κd
2
2
2
2
Evidently, this constraint directly translates to the maximal sequence length n,
i.e.,
n
n+1
3
3
ln h
ln r
≤r<h
⇐⇒ −1 < n + 3 − 3 ≤ 0.
h
(C.3.12)
ν
ν
ln ν
ln ν
By combining (C.3.10) and (C.3.11), we have that
ν 4d n−1
∥F ∥L1 .
∥F · χΩ(h) ∥L1 ≤ 1 −
2
Finally, by expressing n in terms of the parameter h, according to (C.3.12), we
conclude that
∥F · χΩ(h) ∥L1 ≤ Chβ ∥F ∥L1 ∀ 0 < h ≤ 1
for some constants β, C > 0 only dependent on ν (and d).
Remark. By a closer inspection of the above proof, we actually obtain explicit
expressions for the exponent β > 0 in Theorem C.3.1. In fact, for every fixed
2d
0 < ϵ ≤ ν2 , there is an associated multiplicative constant C = C(ϵ) > 0 such
that Theorem C.3.1 is satisfied with exponent
ν 2d 3 −1
· ln
.
β = − ln (1 + ϵ) · 1 −
2
ν
Although we may improve the estimate for β by choosing a smaller ϵ, this comes at
the cost of an enlarged multiplicative constant C, where, at least from the above
proof scheme, C(ϵ) → ∞ as ϵ → 0.
C.3.2
Fractal uncertainty principle for modulation spaces
By the norm correspondence (C.2.6), we can rephrase Theorem C.3.1 in terms of
the STFT on modulation spaces.
Theorem C.3.5. (FUP for modulation spaces M p (Rd )) Let 0 < h ≤ 1, and
suppose Ω(h) ⊆ R2d is an h-dependent family of sets which is ν-porous on scales
h to 1. Then for all p ≥ 1 and all f ∈ M p (Rd ) \ {0} there exist constants C, β > 0
only dependent on ν (and d) such that
∥Vϕ0 f · χΩ(h) ∥pLp (R2d )
∥Vϕ0 f ∥pLp (R2d )
90
≤ Chβ ∀ 0 < h ≤ 1.
(C.3.13)
C.4. Density of Cantor sets
ϕ0
In particular, for p = 2, Daubechies’ time-frequency localization operator PΩ(h)
satisfies
ϕ0
∥op ≤ Chβ ∀ 0 < h ≤ 1.
∥PΩ(h)
(C.3.14)
Another noteworthy rephrasement is result (C.3.6), which in terms of the STFT
reads
∥Vϕ0 f · χΩ ∥pLp (R2d )
≤ 2−d
1−e
ρ(Ω, R)
∥Vϕ0 f ∥pLp (R2d ) ∀ R > 0.
Pd−1 ( 12 πR2 )j
− 12 πR2
j=0
(C.3.15)
j!
Remark. For d = 1, in [7] Theorem 3, there is a similar but more general result,
valid for all Hermite windows {hj }j , namely
∥Vhj f · χΩ ∥pLp (R2 ) ≤
ρ(Ω, R)
∥Vhj f ∥pLp (R2 ) ∀ p ≥ 1 and R > 0,
1 − e−πR2 Pj (πR2 )
where Pj is a specified polynomial of degree 2j. In particular, for j = 0, we have
that h0 = ϕ0 and P0 ≡ 1 so that the above inequality reduces to
∥Vϕ0 f · χΩ ∥pLp (R2 ) ≤
ρ(Ω, R)
∥Vϕ0 f ∥pLp (R2 ) ∀ p ≥ 1 and R > 0.
1 − e−πR2
(C.3.16)
By close inspection, for d = 1 result (C.3.15) turns out to be an improvement of
(C.3.16).2
C.4
Density of Cantor sets
In this section we consider the Cantor sets specifically and show, under certain
conditions, how the FUP in Theorem C.3.5 (or equivalently Theorem C.3.1) can
be refined for this family of fractal sets. While the general FUP is formulated in
terms of a continuous parameter h → 0+ , it is more convenient to consider discrete
iterations n → ∞ when working with the Cantor sets. The relation between h and
n is made clear by Lemma C.2.1, where for the Cantor iterate Cn (M, A , L), we
have that
h ∼ LM −n .
From the above relation, we observe that h is of the same order of magnitude as the
intervals that make up the Cantor iterate. Furthermore, similarly to the condition
imposed in [27], we consider L ∼ h−1 . Thus, we have that L is dependent on
n
the iterates n so that L(n) ∼ M 2 . More precisely, we specify the multiplicative
constants of the asymptotes so that the interval condition now reads:
2 In [62] (Section 3.2 page 10) there is a similar comparison for a specific set Ω, where ρ(Ω, R) is
known, with the claim that the upper bound obtained in [62] is also an improvement of (C.3.16).
By closer examination, this claim is incorrect, and the upper bound for the localization operator
is in fact the same as result (C.3.16), i.e., the special case h0 = ϕ0 of Theorem 3 in [7].
91
Paper C. An FUP for the STFT and Gabor multipliers
Def. C.4.1. Let the interval length L be a function N → R+ . The interval length
L satisfies condition (IM ) with constants 0 < c1 ≤ c2 < ∞ if
n
n
c1 M 2 ≤ L(n) ≤ c2 M 2 for n = 0, 1, 2, . . .
(C.4.1)
For the 2d-dimensional radially symmetric Cantor iterate Cn2d (R, M, A ), we
adjust the above condition to the radius R > 0, similar to condition (1.3) in [62]
and (1.4) in [63].
Def. C.4.2. Let the radius R be a function N → R+ . The radius R satisfies
condition (DM ) with constants 0 < c1 ≤ c2 < ∞ if
n
n
c1 M 2 ≤ R2d (n) ≤ c2 M 2 , for n = 0, 1, 2, . . .
(C.4.2)
With these conditions, we present an FUP for the two multidimensional Cantor
set constructions from Section C.2.4.
Theorem C.4.1. (FUP for Cantor sets) We consider the two constructions separately:
(i) Let Cn2d (R, M, A ) denote the 2d-dimensional radially symmetric Cantor iterate, and suppose the radius R = R(n) > 0 satisfies condition (DM ) with
constants c1 ≤ c2 . Then there exists a positive constant γ only dependent
on M , |A |, c1 , c2 (and d) such that for all f ∈ M p (Rd ) with p ≥ 1 and all
iterations n ∈ N0 ,
∥Vϕ0 f ·
χCn2d (R(n),M,A ) ∥pLp (R2d )
≤γ·
|A |
M
n2
∥Vϕ0 f ∥pLp (R2d ) .
(C.4.3)
(ii) Let Ω := Cn1 (L1 , M1 , A1 )×· · ·×Cn2d (L2d , M2d , A2d ) denote the 2d-dimensional
Cartesian product of Cantor iterates, and suppose the initial interval lengths
Lj = Lj (nj ) satisfy condition (IMj ) with the same constants c1 ≤ c2 . Then
there exists a positive constant γ only dependent on {(Mj , |Aj |)}j , c1 , c2 (and
d) such that for all f ∈ M p (Rd ) with p ≥ 1 and all iterations {nj }j ∈ N2d
0 ,


nj /2
2d Y
|A
|
j
 ∥Vϕ0 f ∥p p 2d .
∥Vϕ0 f · χΩ ∥pLp (R2d ) ≤ γ · 
L (R )
M
j
j=1
In particular, when all iterates are equal to say Cn (L(n), M, A ), we find that
∥Vϕ0 f ·
χCn (L,M,A )2d ∥pLp (R2d )
≤γ·
|A |
M
n·d
∥Vϕ0 f ∥pLp (R2d ) .
(C.4.4)
Remark. For localization on the radially symmetric Cantor set with d = 1 and
p = 2, we retrieve the same upper bound asymptote as in [63] Theorem 3.3. In
fact, Theorem 3.3 reveals that the general asymptote is optimal in the sense that
there are alphabets where the asymptote is precise. However, the same theorem
also states that there are alphabets where the asymptote is not precise.
92
C.4. Density of Cantor sets
The proof of Theorem C.4.1 is based on inequality (C.3.15), combined with
estimates of the maximal Nyquist density of the multidimensional Cantor iterates.
In order to produce such estimates, we first need to consider the 1-dimensional
Cantor iterates and introduce the notion of the Cantor function. To the n-iterate
Cantor set Cn (M, A ) := Cn (1, M, A ) based in [0, 1] we associate the Cantor function Gn,M,A : R → [0, 1], given by
(
−1 0, x ≤ 0,
(C.4.5)
Gn,M,A (x) := |Cn (M, A )|
|Cn (M, A ) ∩ [0, x]|, x > 0.
For the iterates based in the interval [0, L], we consider the dilated Cantor function Gn,M,A (· L−1 ). The Cantor function is a useful concept as the difference
Gn,M,A (y) − Gn,M,A (x) measures the portion of the n-iterate Cn (M, A ) contained
in the interval [x, y]. By definition, the Cantor function is said to be subadditive
if the difference is bounded by Gn,M,A (y − x). While this is the case for the midthird Cantor set (see [24]), we cannot guarantee subadditivity with an arbitrary
alphabet. Nonetheless, for our purpose, we only require a weaker version utilizing
the canonical alphabet A := {0, 1, . . . , |A | − 1} (see [63] Appendix A).
Lemma C.4.2. Let Gn,M,A denote the Cantor-function, defined in (C.4.5). Then
for any x ≤ y,
Gn,M,A (y) − Gn,M,A (x) ≤ Gn,M,A (y − x).
(C.4.6)
Re-scaling to Cantor iterates based in [0, L], we find, by the above lemma, that
the maximal Nyquist density is bounded by
ρ Cn (L, M, A ), x = max |Cn (L, M, A ) ∩ [a, a + x]| ≤ |Cn (L, M, A ) ∩ [0, x]|.
a∈R
In the next lemma, we enforce condition (C.4.1), which yields a more explicit
estimate for the upper bound in the 1-dimensional case.
Lemma C.4.3. Suppose that the alphabet A is a proper subset of {0, 1, . . . , M −1},
and suppose that the length L = L(n) > 0 satisfies condition (IM ) with constants
0 < c1 ≤ c2 < ∞. Then for any fixed x > 0, there exists a finite constant γ > 0
dependent only on x, c1 , c2 such that
|Cn (L, M, A ) ∩ [0, x]| = Gn,M,A xL
−1
|A |
M
n
L≤γ·
|A |
M
n2
.
(C.4.7)
Proof. Fix some positive integer n0 < n, and observe that for coefficients aj ∈ A ,
Pn−1
the sum j=0 aj M j−n < M n0 −n only if aj = 0 for j ≥ n0 . This means that
(d)
whenever x < M n0 −n , the cardinality of the intersection Cn (M, A ) · M −n ∩ [0, x]
cannot exceed |A |n0 . We also note that the discrete n-iterate Cantor set itself has
cardinality |A |n . Thus, the associated Cantor function must be bounded by
Gn,M,A (x) ≤ |A |−m ∀ x ≤ M −m and m = 0, 1, . . . , n − 1.
93
Paper C. An FUP for the STFT and Gabor multipliers
1
Since the same upper bound holds if we suppose x ≤ M −(m+ 2 ) < M −m , it follows
that
1
m
m
Gn,M,A (x) ≤ |A | 2 |A |− 2 ∀ x ≤ M − 2 and m = 0, 1, . . . , 2(n − 1).
Now, let x be any positive, fixed number and consider the argument xL−1 . Utin
lizing the lower bound condition L(n) ≥ c1 M 2 , we can find a positive integer
N
−n
−1
N = N (xc−1
≤ M 2 . By monotonicity of the Cantor function,
1 ) such that xL
combined with the above estimate for the Cantor function, we conclude
N −n N −n
1
Gn,M,A xL−1 ≤ Gn,M,A M 2
≤ |A | 2 |A | 2 ∀ n = 0, 1, 2, . . .
n
Finally, we enforce the upper bound condition L(n) ≤ c2 M 2 , and inequality
N +1
(C.4.7) follows with constant γ = c2 |A | 2 . Since 0 < c1 ≤ c2 < ∞, the constant
γ is indeed finite.
Based on the 1-dimensional result, we estimate upper bounds for the maximal
Nyquist densities for the multidimensional Cantor iterates, which, combined with
(C.3.15), proves Theorem C.4.1.
Lemma C.4.4. (Nyquist density of Cantor sets) Let r > 0 be fixed.
(i) Let Cn2d (R, M, A ) denote the radial Cantor iterate, and suppose the radius
R = R(n) > 0 satisfies condition (DM ) with constants c1 ≤ c2 . Then there
exists a constant γr > 0 only dependent on M, |A |, c1 , c2 (and d) such that
the maximal Nyquist density is bounded by
ρ Cn2d (R, M, A ), r ≤ γr ·
|A |
M
n2
for n = 0, 1, 2, . . .
(ii) Let Ω := Cn1 (L1 , M1 , A1 ) × · · · × Cn2d (L2d , M2d , A2d ) denote the Cartesian
product, and suppose the interval lengths Lj = Lj (nj ) satisfy condition (IMj )
with the same constants c1 ≤ c2 . Then there exist a constant γr > 0 only
dependent on {(Mj , |Aj |)}j , c1 , c2 (and d) such that the maximal Nyquist
density is bounded by
ρ (Ω, r) ≤ γr ·
n /2
2d Y
|Aj | j
j=1
Mj
for nj = 0, 1, 2, . . .
Proof. Without loss of generality, we can assume that r ≥ 1. For part (i), fix
a cutoff value N ≫ 1 and distinguish between two cases for Br (a) ⊆ R2d , (1)
|a| ≤ N r and (2) |a| > N r:
For case (1), Br (a) ⊆ B(N +1)r (0) so that
|Cn2d (. . . ) ∩ Br (a)| ≤ |Cn2d (. . . ) ∩ B(N +1)r (0)|.
94
C.4. Density of Cantor sets
By definition of the radial Cantor iterate, it follows that
πd
Cn (R2d , M, A ) ∩ 0, (N + 1)2d r2d
d!
n
|A | 2
≤ γ1 ·
(by Lemma C.4.3)
M
Cn2d (R, M, A ) ∩ Br (a) ≤
for some constant γ1 > 0.
For case (2), Br (a) ⊆ Ar (|a|) := {z ∈ R2d | 0 < |a| − r ≤ |z| ≤ |a| + r}. For
the annulus Ar (|a|) with r < |a|, we have again, by definition of the radial Cantor
iterate, that
πd
Cn (R2d , M, A ) ∩ (|a| − r)2d , (|a| + r)2d
d!
πd
≤
(by Lemma C.4.2).
Cn (R2d , M, A ) ∩ 0, (|a| + r)2d − (|a| − r)2d
d!
(C.4.8)
Cn2d (R, M, A ) ∩ Ar (|a|) =
Since |a| > 1 and since the leading term |a|2d cancels, it follows that
(|a| + r)2d − (|a| − r)2d ≤ |a|2d−1 (1 + r)2d − (1 − r)2d .
Furthermore, by subadditivity, the Cantor function Gn,M,A (m·x) ≤ m· Gn,M,A (x)
for any m ∈ N. In general, this means
Gn,M,A (m · x) ≤ (m + 1) · Gn,M,A (x) ∀ m > 0.
With these observations inequality (C.4.8) simplifies to
Cn2d (R, M, A ) ∩ Ar (|a|)
πd
|a|2d−1 + 1 Cn (R2d , M, A ) ∩ 0, (1 + r)2d − (1 − r)2d
≤
d!
n
|A | 2
2d−1
(by Lemma C.4.3)
≤ γ2 · |a|
+1
M
(C.4.9)
for some constant γ2 > 0 independent of |a|. Evidently, the right-hand side of
the above inequality is unbounded in terms of |a|. The ball Br (a), however, only
represents a fraction of the annulus Ar (|a|), which warrants a closer comparison
between |Cn2d (. . . )∩Ar (|a|)| and |Cn2d (. . . )∩Br (a)|. Let ∂Bη := {z ∈ R2d | |z| = η}
denote the (2d−1)-dimensional sphere of radius η > 0 with associated surface area
|∂Bη |. We consider the optimal surface quotient
max
|∂Bη ∩ Br (a)|
|∂Bη |
2d−1
r
η ∈ |a| − r, |a| + r ≤ α ·
|a|
for some constant α > 0 only dependent on N ≫ 1 and d. Hence,
2d−1
r
Cn2d (R, M, A ) ∩ Br (a) ≤ α ·
Cn2d (R, M, A ) ∩ Ar (|a|) ,
|a|
95
Paper C. An FUP for the STFT and Gabor multipliers
which, combined with (C.4.9), yields the desired conclusion of part (i).
Part (ii) turns out to be much simpler. Since any ball Br (a) is contained in
some shifted hypercube a + [−r, r]2d , we consider the maximal Nyquist density
ρ Cnj (Lj , Mj , Aj ), r) in each direction j = 1, 2, . . . , 2d. In total, we obtain
ρ(Ω, r) ≤
2d
Y
ρ Cnj (Lj , Mj , Aj ), r),
j=1
where the desired conclusion follows once we apply the 1-dimensional result from
Lemma C.4.3.
C.5
Fractal uncertainty principle for Gabor
multipliers
In this section we present one simple translation of Theorem C.3.1 to Gabor multipliers. Specifically, since the previous theorems are based on the Gaussian window
ϕ0 , we proceed with Gabor multipliers on the form (C.2.10) with ϕ0 as generating
function, i.e.,
X
ϕ0
GΛ,Ω
f = |DΛ |
⟨f, π(λ)ϕ0 ⟩π(λ)ϕ0
(C.5.1)
λ∈ΛΩ
2d
for some subset Ω ⊆ R and lattice Λ. Furthermore, since we consider an hdependent family Ω(h) of sets for the FUP, we also let the lattice Λ depend on the
parameter 0 < h ≤ 1, meaning, we let Λ(h) become sufficiently dense so to capture
the fractal details of Ω(h). In particular, we consider the following condition:
Def. C.5.1. We say that an h-dependent family of lattices Λ(h) ⊆ R2d for 0 <
h ≤ 1 satisfies condition (H) with constant L > 0 if the fundamental region, DΛ(h) ,
satisfies the inclusion
DΛ(h) ⊆ BhL (0) ∀ 0 < h ≤ 1.
Remark. Any family of rectangular lattices of the form Λ1 (h) = h aZd × bZd
p
with a, b > 0 satisfies condition (H)
L = 2−1 d(a2 + b2 ) > 0. In
dwith constant
2
d
contrast, the family Λ2 (h) = h a Z × bZ does not satisfy condition (H), even
though |DΛ1 (h) | = |DΛ2 (h) |.
Utilizing condition (H), we formulate the FUP for Gaussian Gabor multipliers:
Theorem C.5.1. (FUP Gaussian Gabor multipliers) Let 0 < h ≤ 1, and suppose
Ω(h) ⊆ R2d is an h-dependent family of sets which is ν-porous on scales h to 1.
ϕ0
Let GΛ(h),Ω(h)
: L2 (Rd ) → L2 (Rd ) denote the Gaussian Gabor multiplier defined in
(C.5.1), whose lattice Λ(h) satisfies condition (H) with constant L > 0. Then there
exists constants C, β > 0 only dependent on ν, L (and d) such that the operator
norm is bounded by
ϕ0
∥op ≤ Chβ ∀ 0 < h ≤ 1.
∥GΛ(h),Ω(h)
96
C.5. Fractal uncertainty principle for Gabor multipliers
First, we reformulate the problem to an estimate in the Fock space.
Lemma C.5.2. Let Ω∗ := {z ∈ Cd | z ∈ Ω} denote the complex conjugate set of
ϕ0
Ω ⊆ R2d . The operator norm of the Gaussian Gabor multiplier GΛ,Ω
: L2 (Rd ) →
2
d
L (R ) is given by
ϕ0
∥GΛ,Ω
∥op =
|DΛ |
sup
F ∈F 2 (Cd ), ∥F ∥L2 =1
2
X
|F (λ)|2 e−π|λ| .
λ∈ΛΩ∗
Proof. By Cauchy-Schwarz’ inequality on ℓ2 -sequences, the operator norm is given
by
X
ϕ0
ϕ0
∥GΛ,Ω
∥op = sup ⟨GΛ,Ω
f, f ⟩ = sup |DΛ |
|⟨f, π(λ)ϕ0 ⟩|2 .
∥f ∥2 =1
∥f ∥2 =1
λ∈ΛΩ
By identity (C.2.5), each term can be expressed in terms of the Bargmann trans2
form, namely |⟨f, π(λ)ϕ0 ⟩|2 = |B f (λ)|2 e−π|λ| , and since the Bargmann transform
B : L2 (Rd ) → F 2 (Cd ) is an isometry onto the Fock space, the result follows.
Proof. (Theorem C.5.1) By Lemma C.5.2, we equivalently consider the following
P
2
sum |DΛ | λ |F (λ)|2 e−π|λ| for any normalized F ∈ F 2 (Cd ). By the subaveraging
property in Lemma C.3.2, for every R > 0

X
2
|F (λ)|2 e−π|λ| ≤ 1 − e−πR
d−1
2 j
2 X (πR )
j=0
λ

≤ 1 − e−πR
2
2
|F (ξ)|2 e−π|ξ| dA(ξ)
BR (λ)
λ
d−1
X
(πR2 )j
j=0
XZ

j!
j!
−1
−1

γ(R, Λ)
Z
2
|F (ξ)|2 e−π|ξ| dA(ξ),
·
S
λ
BR (λ)
where
γ(R, Λ) := sup card λ ∈ Λ
|B2R (z) ∩ (DΛ + λ)| > 0
z∈DΛ
takes into account the possible overlap between the balls BR (λ). We are interested
in the case R = h for the family of lattices Λ(h). By the condition DΛ(h) ⊆ BLh (0),
it follows that
|B2h (z) ∩ (DΛ(h) + λ)| > 0 =⇒ |B(2+L)h (z) ∩ (DΛ(h) + λ)| = |DΛ(h) |.
In turn, the overlap must be bounded by
γ(h, Λ(h)) ≤
|B(2+L)h (0)|
|Bh (0)|
= (2 + L)2d ·
∀ 0 < h ≤ 1.
|DΛ(h) |
|DΛ(h) |
97
Paper C. An FUP for the STFT and Gabor multipliers
In total, we obtain
X
2
|DΛ(h) |
|F (λ)|2 e−π|λ| ≤ (2+L)2d · κd (πh2 )
λ∈ΛΩ∗ (h)
Z
2
|F (ξ)|2 e−π|ξ| dA(ξ),
·
S
λ∈ΛΩ∗ (h)
Bh (λ)
where κd (x) was defined in (C.3.8). Since κd (x) is continuous for x > 0 and
limx→0 κd (x) = 1, this factor is simply absorbed by the multiplicative constant
C > 0 of the theorem.
∗
Now, by definition
S (C.2.9) for the subset ΛΩ (h) ⊆ Λ(h) and again by DΛ(h) ⊆
BhL (0), the union λ∈ΛΩ∗ (h) Bh (λ) must be contained in the r-thickened set Ω∗r
for r(h) := (1 + L)h. By Lemma C.3.4, the thickened set Ω∗r(h) is itself ν2 -porous
on scales h1 := ν2 r(h) to 1. Thus, we simply apply the FUP for the Fock space in
Theorem C.3.1 on the family of sets Ω∗r(h) as h1 → 0, from which the statement
follows.
Remark. The question that immediately arises is whether the Gabor systems
{π(λ)ϕ0 }λ∈Λ(h) with Λ(h) satisfying condition (H) actually are Gabor frames for
all 0 < h ≤ 1 or for sufficiently small h. In d = 1, by the results obtained
independently by Lyubarskii [72] and Seip and Wallsten [76], [77], we have a simple
density criterion. Namely, that the system {π(λ)ϕ0 }λ∈Λ is a Gabor frame if and
only if the fundamental region satisfies |DΛ | < 1. In higher dimensions d > 1,
characterizations of lattices Λ that yield Gaussian Gabor frames becomes much
more intricate, which have been studied in [49], [74], [51], [71]. In particular, the
density criterion does not translate, e.g., as shown in [71] Theorem 1.5, the lattice
Λ = (Z× 21 Z)2 does not generate a Gaussian Gabor frame even though |DΛ | = 41 <
1. Further, a sufficient condition is formulated in [71] Theorem 1.2 combining a
density criterion with the notion of transcendental lattices, which, as remarked by
the authors, represents a large family of lattices in Cd . Nonetheless, our estimates
of the Gabor multiplier are not reliant on the operator being associated to a Gabor
frame.
We conclude this section with a simple example of Gabor multipliers based on
Cantor sets, which also illustrates an alternative approach to choosing the lattice
restriction ΛΩ .
Example C.5.1. (Cantor set) For simplicity, we let d = 1 and consider the symmetric Cartesian product of Cantor iterates Ω := Cn (L, M, A ) × Cn (L, M, A ). For
this case one obvious choice of lattices are square lattices Λ = a (Z × Z) with density a ∼ LM −n . Unsurprisingly, the restriction ΛΩ closely resembles the Cartesian
(d)
product of scaled discrete Cantor iterates Ω(d) := [LM −n · Cn (M, A )]2 . Thus,
for Gabor multipliers localizing on such Cartesian products, it seems more natural
to consider sampled points directly from the already available discrete set, i.e., we
consider the operator
X
Gϕ0 (Ω(d) )f := (LM −n )2
⟨f, π(λ)ϕ0 ⟩π(λ)ϕ0 .
2
λ∈[LM −n ·C (d) (M,A )]
98
C.5. Fractal uncertainty principle for Gabor multipliers
In order to estimate the operator norm, we follow the same procedure as in the
proof of Theorem C.5.1. After one 12 LM −n -thickening and then utilizing inequality
(C.3.15), we obtain for all f ∈ L2 (R) and r > 0 that
4 κ1 π4 (LM −n )2 (d)
(d)
1
∥f ∥22 .
∥Gϕ0 (Ω )f ∥2 ≤
ρ
Ω
+
B
−n (0), r
1
LM
2
2
π
1 − e 2 πr
By definition of the discrete and ”continuous” Cantor iterate, the Nyquist density
must satisfy ρ(Ω(d) + B 21 LM −n (0), r) ≤ ρ(Ω, r), so the operator norm is in turn
bounded by
4 κ1 π4 (LM −n )2
∥Gϕ0 (Ω(d) )∥op ≤
ρ(Ω, r).
1
2
π
1 − e 2 πr
If we now suppose the length L depends on the iterates n according to condition
(IM ), we simply apply the estimate for the Nyquist density in Lemma C.4.4 to
retrieve the same asymptotic estimate for the operator norm as in Theorem C.4.1.
Acknowledgements
The research of the author was supported by Grant 275113 of the Research Council
of Norway. The author would like to extend thanks to prof. Eugenia Malinnikova
for many insightful discussions that helped shape the ideas in the manuscript.
The author would also like to thank prof. Franz Luef for the support and who
suggested that fractal uncertainty principles for the Daubechies’ operator could
be translated to the context of Gabor multipliers. Finally, the author is grateful
to the anonymous referees who’s many constructive comments helped improve an
earlier version of the manuscript.
99
Paper C. An FUP for the STFT and Gabor multipliers
100
Appendix
C.A
Complex interpolation in Fock space
We prove the following inequality between the Fock spaces F 1 (Cd ) and F p (Cd )
for p ≥ 1.
Lemma C.A.1. For any measurable subset Ω ⊆ Cd and any p ≥ 1, we have the
inequality
∥F · χΩ ∥pLp
∥F · χΩ ∥L1
≤
sup
.
p
∥F
∥
∥F ∥L1
p
d
1
d
p
F ∈F (C )\{0}
F ∈F (C )\{0}
L
sup
The proof combines central results from complex interpolation. An introduction
to complex interpolation, based on Hadamard’s three line theorem, can be found in
[86] Chapter 2. We follow the notation in [86] and let Xθ = [X0 , X1 ]θ for θ ∈ [0, 1]
denote the complex interpolation space between two (compatible) Banach spaces
X0 and X1 . To begin with, we present a general result regarding linear operators
between interpolation spaces and the associated operator norm (from [86] Theorem
2.4 (c) combined with subsequent remark).
Theorem C.A.2. Suppose X0 , X1 and Y0 , Y1 are compatible pairs of Banach
spaces, and suppose the mapping
T : X0 + X1 → Y0 + Y1
is bounded, linear such that T : Xk → Yk is bounded with norm ∥T ∥k ≤ Mk
for k = 0, 1. Then the mapping T satisfies T : [X0 , X1 ]θ → [Y0 , Y1 ]θ with norm
estimate ∥T ∥θ ≤ M01−θ M1θ for all θ ∈ [0, 1].
In order to relate the above theorem to our context, we need to consider interpolation between weighted Lp -spaces and between Fock spaces. The next statements are all found in [87] Chapter 2.4, formulated for spaces over C but easily
generalized to Cd . First, we consider the Stein-Weiss interpolation theorem, first
published in [80], for weighted Lp -spaces.
Theorem C.A.3. Suppose w, w0 and w1 are positive weight functions on Cd .
Then for any 1 ≤ p0 ≤ p1 ≤ ∞ and θ ∈ [0, 1], we have that
p0 d
L (C , w0 dA), Lp1 (Cd , w1 dA) θ = Lp (Cd , wdA),
where
1−θ
θ
1
1
1−θ
θ
=
+
and w p = w0p0 w1p1 .
p
p0
p1
In particular, for the Lp -spaces with Gaussian measures, Lp (Cd ), we obtain:
Corollary C.A.1. For any 1 ≤ p0 ≤ p1 ≤ ∞ and θ ∈ [0, 1], we have that
Lp0 (Cd ), Lp1 (Cd )
θ
= Lp (Cd ), where
101
1
1−θ
θ
=
+ .
p
p0
p1
Paper C. An FUP for the STFT and Gabor multipliers
For Fock spaces we have the following interpolation:
Theorem C.A.4. For any 1 ≤ p0 ≤ p1 ≤ ∞ and θ ∈ [0, 1], we have that
p0 d
1
1−θ
θ
F (C ), F p1 (Cd ) θ = F p (Cd ), where
=
+ .
p
p0
p1
With these interpolation results, we are ready to prove Lemma C.A.1.
Proof. (Lemma C.A.1) Note that the statement is equivalent to the linear mapping
TΩ : F p (Cd ) → Lp (Cd ), F 7→ F · χΩ
satisfying the operator norm inequality ∥TΩ ∥pLp ≤ ∥TΩ ∥L1 for p ≥ 1. Therefore,
we consider the mapping TΩ between the spaces
TΩ : F 1 (Cd ) + F ∞ (Cd ) → L1 (Cd ) + L∞ (Cd ),
which is clearly bounded. By Theorem C.A.2,
TΩ : F 1 (Cd ), F ∞ (Cd ) θ → L1 (Cd ), L∞ (Cd ) θ ,
θ
and the associated operator norm is bounded by ∥TΩ ∥1−θ
L1 ∥TΩ ∥L∞ . By Corollary
C.A.1 and Theorem C.A.4, these interpolation spaces correspond to F p (Cd ) and
Lp (Cd ), respectively, with p−1 = 1 − θ. In addition, ∥TΩ ∥L∞ ≤ 1, from which the
norm estimate readily follows.
C.B
Omitted proof: Simple porosity estimate of
Cantor sets
We shall prove the following simple porosity estimate for the n-iterate Cantor set
in 1-dimension.
Lemma C.B.1. The n-iterate Cantor set Cn (L, M, A ) with base M > 1 and
alphabet size |A | < M , based in the interval [0, L], is ν-porous on scales LM −n+1
to ∞, with any ν ≤ 12 M −2 .
Proof. Without loss of generality, we assume L = 1. Consider an interval I ⊆ R
of size M −m+1 ≤ |I| ≤ M −m+2 for some integer 1 ≤ m ≤ n. Suppose first that
the intersection I ∩ Cm (. . . ) does not form an interval. Then, by the Cantor set
construction, there exists an interval J ⊆ I of size |J| = M −m ≥ M −2 |I| such
that |J ∩ Cm (. . . )| = 0. Conversely, suppose the intersection forms an interval,
effectively dividing the remainder I \ Cm (. . . ) into two intervals J1 , J2 . Again, by
the Cantor set construction, we have the upper bound |I ∩ Cm (. . . )| ≤ |A |M −m ≤
(M −1)M −m , so that max{|J1 |, |J2 |} ≥ 12 (|I|−(M −1)M −m ). Hence, we conclude
that there exists an interval J ⊆ I with |I ∩ Cm (. . . )| = 0 of size
|I| − (M − 1)M −m
1
|J| =
|I| ≥ M −2 |I|.
2|I|
2
Since m is arbitrary and Cn (. . . ) ⊆ Cm (. . . ), the statement holds for all intervals I
with M −n+1 ≤ |I| ≤ M . For |I| ≥ M the statement becomes trivial as Cn (. . . ) ⊆
[0, 1].
102
C.C. Subaveraging in Fock space
C.C
Subaveraging in Fock space
We shall generalize the subaveraging statement made in Lemma 2.32 in [87] for
the Fock space over C to the space over Cd , namely:
Lemma C.C.1. For any F ∈ F p (Cd ) and any point z ∈ Cd , we have for all
R > 0 that

−1
d−1 p
2 j
X
d
p
p
2
2
p 
2 πR

|F (z)|p e− 2 π|z| ≤
1 − e− 2 πR
2
j!
j=0
(C.C.1)
Z
p
2
·
|F (ξ)|p e− 2 π|ξ| dA(ξ).
BR (z)
In one dimension, the proof is based on the subaveraging property of subharmonic functions in C. More precisely, for u : C → R ∪ {−∞} subharmonic, we
have that
Z 2π
1
u(z) ≤
u(reiθ + z)dθ ∀ r > 0.
2π 0
In d-dimensions, we instead consider plurisubharmonic functions.
Def. C.C.1. (Plurisubharmonic) Let X be a domain in ⊆ Cd . We say that a
function u : X → R ∪ {−∞} is plurisubharmonic if
(a) u is upper semi-continuous, and
(b) for every z, ξ ∈ Cd the function τ 7→ u(ξτ + z) is subharmonic in the open
subset of C where it is defined.
Many well-known examples of plurisubharmonic functions are based on the
holomorphic functions. In particular, if F is entire, then |F |p is a plurisubharmonic
function in Cd for every p > 0. Hence, by point (b) in Def. C.C.1, every F ∈
F p (Cd ) satisfies
1
|F (z)| ≤
2π
p
Z
2π
p
F (ξreiθ + z) dθ ∀ ξ ∈ Cd and r > 0.
(C.C.2)
0
Notice that since the left-hand side of (C.C.2) is independent of ξ (and r), we may
multiply both sides by a positive function of ξ, integrate with respect to ξ and
keep the inequality intact.
Introduce the notation
∂B1 := z ∈ Cd |z| = 1
for the (2d−1)-dimensional unit sphere, and let |∂B1 | denote the associated surface
area. Further, let dS denote the surface measure on ∂B1 . To begin with, we show
that the point evaluation |F (z)|2 is bounded by an average over the sphere.
103
Paper C. An FUP for the STFT and Gabor multipliers
Lemma C.C.2. For every entire function F , we have that
Z
1
p
|F (z)|p ≤
F (ξr + z) dS(ξ) ∀ r > 0.
|∂B1 | |ξ|=1
(C.C.3)
Proof. Consider the equivalence relation ∼ in Cd , where
ξ1 ∼ ξ2 ⇐⇒ ∃ θ ∈ [0, 2π] such that ξ1 = eiθ ξ2 .
This induces the quotient space ∂B1 / ∼ of ”remaining angles” over the unit sphere,
with measure dΦ such that dS = dθdΦ and
Z 2π Z
Z
|∂B1 | =
dθ
dΦ = 2π
dΦ.
0
∂B1 /∼
∂B1 /∼
With the above decomposition, we may integrate both sides of (C.C.2) over the
space ∂B1 / ∼, which leaves the desired result.
Proceeding, we relate the subaverage over the sphere to an subaverage over the
ball BR (0).
Lemma C.C.3. For every entire function F , we have for all R > 0 that

−1
d−1 p
p d
2 j
X
πR
p
2
2
1 − e− 2 πR

|F (z)|p ≤
2
j!
j=0
(C.C.4)
Z
p − p π|ξ|2
F (z + ξ) e 2
·
dA(ξ).
BR (0)
Proof. For any ω ∈ Cd , we have the basic decomposition ω = rξ for |ξ| = 1 and
r = |ω|, so that the volume measure can be expressed dA(ω) = r2d−1 drdS(ξ).
Since we consider the Gaussian measure of the Fock space F p (Cd ), we integrate
p
2
both sides of inequality (C.C.3) against e− 2 πr r2d−1 dr. For the right-hand side,
we find that
Z RZ
p
2
1
p
F (ξr + z) e− 2 πr r2d−1 dS(ξ)dr
|∂B1 | 0 |ξ|=1
Z
p
2
1
p
=
F (ω + z) e− 2 π|ω| dA(ω).
|∂B1 | BR (0)
While for the left-hand side, using the formula


Z L
k
j
X
L 
rk e−r dr = k! 1 − e−L
for k = 0, 1, 2, . . . ,
j!
0
j=0
we obtain
|F (z)|p
Z
e
0
104
R
2
−p
2 πr
(d − 1)!
r2d−1 dr = |F (z)|p
2
2
pπ
d

1 − e
2
−p
2 πR
d−1
X
j=0
p
2 j
2 πR
j!

.
C.D. Generalizations of the fractal uncertainty principle
Since the surface area of the (2d − 1)-dimensional unit sphere is given by |∂B1 | =
2π d /(d − 1)!, inequality (C.C.4) follows.
We are now ready to prove Lemma C.C.1.
Proof. (Lemma C.C.1) Consider the integral over the ball centered at z ∈ Cd ,
Z
Z
p
2
p
p −p
π|ξ|2
2
|F (ξ)| e
dA(ξ) =
F (z + ξ) e− 2 π|z+ξ| dA(ξ)
BR (z)
B (0)
Z R
(C.C.5)
p
p
2
2
p
F (z + ξ)e−πξ·z e− 2 π|ξ| dA(ξ).
= e− 2 π|z|
BR (0)
Define Gz (ξ) := F (z +ξ)e−πξ·z , which is an entire function with respect to ξ. Since
Gz (0) = F (z), the inequality follows once we apply Lemma C.C.3 to |Gz (0)|p ,
combined with (C.C.5).
C.D
Generalizations of the fractal uncertainty
principle
In this section we discuss an easy extension of the FUP of Theorem C.3.1 on the
Fock space, i.e., the image of the STFT with the Gaussian window, to a larger
family of window functions. In order to do so, we first identify the essential features
of the Fock space that we utilized in the proof, which turns out was only the
subaveraging property of the Fock space. Subsequently, we ask whether a similar
subaveraging feature is present for function spaces generated by different windows
than the Gaussian.
Originally, when we introduced the modulation spaces M p (Rd ) back in Section
C.2.1, we only considered a fixed window function ϕ ∈ S (Rd ) \ {0}. However, from
Jakobsen’s detailed survey paper [57], with equivalent definitions labeled A to U
for Feichtinger’s algebra M 1 (Rd ), we read (from definition H ) that M 1 (Rd ) and
by extension M p (Rd ) can be defined using windows from M 1 (Rd ) \ {0} itself, with
equivalent norms defined in the natural way. Since it makes sense to consider the
modulation spaces with window function in M 1 (Rd ) \ {0}, this relaxed condition
on the window will later allow us to construct explicit counter-examples where the
STFT is no longer subaveraging.
Note also that, similar to the Fock space, we only need to consider the case
p = 1 as the modulation spaces themselves satisfy the interpolation property (see
Theorem 1.1 in [53])
p0 d
1
1−θ
θ
M (R ), M p1 (Rd ) θ = M p (Rd ), where
=
+ .
p
p0
p1
By a similar argument to that of Appendix C.A, we find that for any fixed window
ϕ ∈ M 1 (Rd ) \ {0}, any measurable subset Ω ⊆ R2d and p ≥ 1,
sup
f ∈M p (Rd )\{0}
∥Vϕ f · χΩ ∥pLp (R2d )
∥Vϕ f ∥pLp (R2d )
≤
sup
f ∈M 1 (Rd )\{0}
∥Vϕ f · χΩ ∥L1 (R2d )
.
∥Vϕ f ∥L1 (R2d )
105
Paper C. An FUP for the STFT and Gabor multipliers
We now make a precise definition of our sought after subaveraging property:
Def. C.D.1. (Subaveraging for the STFT) Suppose ϕ ∈ M 1 (Rd ) \ {0} is such
that for all radii R > 0 there exists a constant C(R) > 0 such that Vϕ satisfies
Z
Vϕ f (z) ≤ C(R)
Vϕ f (ξ) dA(ξ) ∀ f ∈ M 1 (Rd ) and z ∈ R2d .
BR (z)
Then we say that the STFT with window ϕ, Vϕ , is subaveraging in M 1 (Rd ) with
constant C(R). If, in addition, we have that C(R) · |BR (0)| → 1 as R → 0, we call
the subaveraging retractable.
Hence, whenever Vϕ is retractable subaveraging in M 1 (Rd ) with some constant
C(R) > 0, we can retrace the same steps in the proof of Theorem C.3.1 to produce
an FUP on the modulation spaces but with the Gaussian window replaced by ϕ.
Furthermore, we find an FUP for any finite linear combinations of such windows:
Theorem C.D.1. (FUP generalized) Let the window function be given by ϕ :=
PN
1
d
j=0 αj ϕj for some coefficients αj ∈ C and ϕj ∈ M (R ) \ {0} (and N finite).
1
d
Suppose that Vϕj is retractable subaveraging in M (R ) with constant Cj (R) > 0
for j = 0, 1, . . . , N . Let 0 < h ≤ 1, and suppose Ω(h) ⊆ Cd is an h-dependent
family of sets which is ν-porous on scales h to 1. Then for all p ≥ 1 and all
f ∈ M p (Rd ) \ {0} there exist constants C, β > 0 only dependent on ν, d (and the
window ϕ) such that
∥Vϕ f · χΩ(h) ∥pLp (R2d )
∥Vϕ f ∥pLp (R2d )
≤ Chβ ∀ 0 < h ≤ 1.
ϕ
In particular, for p = 2, Daubechies’ time-frequency localization operator PΩ(h)
satisfies
ϕ
∥PΩ(h)
∥op ≤ Chβ ∀ 0 < h ≤ 1.
Proof. We show the result for the case p = 1, where p ≥ 1 follows as usual
by interpolation. We first use that the STFT is anti-linear with respect to the
window so that
∥Vϕ f · χΩ(h) ∥L1 (R2d ) ≤
N
X
|αj | · ∥Vϕj f · χΩ(h) ∥L1 (R2d ) .
j=0
Since Vϕj is retractable subaveraging for each j = 0, 1, . . . , N , there is an FUP associated with each Vϕj , meaning, there exist constants C (j) , β (j) > 0 only dependent
on ν, d and ϕj such that
∥Vϕj f · χΩ(h) ∥L1 (R2d ) ≤ C (j) hβ
(j)
∥Vϕj f ∥L1 (R2d ) ∀ 0 < h ≤ 1.
By equivalence of norms in M 1 (Rd ), there are also constants D(j) > 0 such that
∥Vϕj f ∥L1 (R2d ) ≤ D(j) ∥Vϕ f ∥L1 (R2d ) ∀ f ∈ M 1 (Rd ).
106
C.D. Generalizations of the fractal uncertainty principle
Combining the above inequalities, we conclude that


N
X
(j)
∥Vϕ f · χΩ(h) ∥L1 (R2d ) ≤ 
|αj |C (j) D(j)  hminj β ·∥Vϕ f ∥L1 (R2d )
j=0
=: Chβ ·∥Vϕ f ∥L1 (R2d ) .
Finally, the assumption that N is finite, ensures that both constants are welldefined, i.e., 0 < C, β < ∞.
We conclude this discussion, with two examples: We first consider a wellknown family of 1-dimensional Schwartz windows, namely the Hermite windows,
whose STFT turns out to be retractable subaveraging. In the second example, we
construct an explicit window function, where the associated STFT is no longer
subaveraging. Note that since the retractable subaveraging property, at this stage,
only represents a sufficient condition for the FUP, the counter-example itself does
not disprove some universal existence of an FUP. Rather, the counter-example
shows that the technique presented in the proof of Theorem C.3.1 is not always
valid, and would have to be altered.
Example C.D.1. (Hermite functions) The Hermite functions {hj }j are given by
j
j
2 d
2
21/4
−1
√
hj (t) = √
eπt j e−2πt for j = 0, 1, 2, . . . ,
dt
j! 2 π
which form a well-known orthonormal basis for L2 (R), and as previously noted, h0
coincides with the Gaussian ϕ0 . Similarly to the Gaussian, the image of the STFT
with a Hermite window. Vhj , also has an analytic structure in terms of (true)
polyanalytic Fock spaces (see [82], [2], [54], [55]). Notably, for p = 2, the image
Vhj (L2 (R)) forms a reproducing kernel Hilbert space with reproducing kernel
π
2
Khj (z, ξ) = ⟨Mω Tx hj , Mη Ty hj ⟩ = eiπ(x+y)·(ω−η) Lj π|z − ξ|2 e− 2 |z−ξ|
for z = x + iω and ξ = y + iη, where Lj denotes the Laguerre polynomial
Lj (x) :=
j
X
(−1)k
k=0
k
j x
.
k k!
This analytic structure has been explored by Abreu and Speckbacher in their
aforemetioned paper [7], where, as a precursor to their large sieve estimates, they
derive local reproducing formulas for Vhj (L2 (R)).
Theorem C.D.2. (Theorem 1 in [7]) For every R > 0 and every j, k ∈ N0 , we
have that
Z
−1
Vhj f (z) = Cj,k (R)
Vhj f (ξ) · Khk (z, ξ) dA(ξ) ∀ f ∈ L2 (R),
BR (z)
with
Z
Cj,k (R) := ⟨χBR (0) · Vhj hj , Vhk hk ⟩ =
πR2
Lj (t)Lk (t)e−t dt.
0
107
Paper C. An FUP for the STFT and Gabor multipliers
Since Khj is given by an inner product of normalized functions, it follows by
Cauchy-Schwarz that the reproducing kernel is bounded by |Kj (z, ξ)| ≤ 1. In
addition, the Laguerre polynomials Lj (x) → 1 as x → 0 for all j (by definition),
which means Cj,k (R) · πR2 → 1 as R → 0. Thus, formulating in terms of M 1 (R)
(since M 1 (R) ⊆ L2 (R); see, e.g., Jakobsen’s survey [57] definition B), and keeping
the same notation as in Theorem C.D.2, the subaveraging property of Vhj reads:
Corollary C.D.1. The STFT with Hermite window hj , Vhj , is retractable subaveraging in M 1 (R) with constant |Cj,k (R)|−1 for any k = 0, 1, 2, . . .
Consequently, there is an FUP for any finite linear combination of Hermite
windows:
Theorem C.D.3. (FUP for modulation spaces M p (R) with Hermite windows)
PN
Let the window function be given by ϕ := j=0 αj hj for some coefficients αj ∈ C
and Hermite functions hj (and N finite). Let 0 < h ≤ 1, and suppose Ω(h) ⊆ C
is an h-dependent family of sets which is ν-porous on scales h to 1. Then for all
p ≥ 1 and all f ∈ M p (R) \ {0} there exist constants C, β > 0 only dependent on ν
(and the window ϕ) such that
∥Vϕ f · χΩ(h) ∥pLp (R2 )
∥Vϕ f ∥pLp (R2 )
≤ Chβ ∀ 0 < h ≤ 1.
ϕ
In particular, for p = 2, Daubechies’ time-frequency localization operator PΩ(h)
satisfies
ϕ
∥PΩ(h)
∥op ≤ Chβ ∀ 0 < h ≤ 1.
Example C.D.2. (Counter–example, where Vϕ is not subaveraging) To begin
with, consider the standard hat function


1 + t, t ∈ [−1, 0],
g(t) := χ[− 1 , 1 ] ∗ χ[− 1 , 1 ] (t) = 1 − t, t ∈ [0, 1],
2 2
2 2


0, otherwise,
and define for j ∈ N0 the dilations
gj (t) := 2−j/2 g(2j t) = 2j/2 χ[−2−(j+1) ,2−(j+1) ] ∗ χ[−2−(j+1) ,2−(j+1) ] (t).
These dilations are constructed such that the L∞ -norm and L1 -norm both tend
to zero but at different rates for increasing j. In particular,
3
∥gj ∥∞ = gj (0) = 2−j/2 and ∥gj ∥1 = 2− 2 j =⇒
∥gj ∥∞
= 2j .
∥gj ∥1
(C.D.1)
Subsequently, we consider the following sum of shifted hat functions
ψ(t) :=
∞
X
j=0
108
gj (t − 2j) =
∞
X
j=0
T2j gj (t).
(C.D.2)
C.D. Generalizations of the fractal uncertainty principle
Since the dilated hats in the above sum are all disjoint, the projection
χ[2j−R,2j+R] · ψ = T2j gj for 0 < R ≤ 1
(C.D.3)
obviously maintains the norm properties of gj . The function ψ will be our candidate for a window function whose STFT is not subaveraging according to Def.
C.D.1.
We first verify that ψ actually belongs to the modulation space M 1 (R). Again,
we refer to one of Jakobsen’s many defintions for M 1 (R) in [57], particularly definition M . This definition involves the Fourier algebra A(R), which is given by
n
o
A(R) := f ∈ C0 (R) ∃ h ∈ L1 (R) such that b
h=f ,
where b
h denotes the Fourier transform of h. Notably the Fourier algebra becomes
a Banach space under the norm ∥f ∥A(R) := ∥h∥1 , and for some fixed K > 0, we
have that
∞
n
X
M 1 (R) = f ∈ L1 (R) f =
Txn fn , where xn ∈ R,
n=0
suppfn ⊆[−K, K], fn ∈ A(R) and
∞
X
(C.D.4)
∥fn ∥A(R)
o
<∞ .
n=0
Lemma C.D.4. Let ψ be defined as in (C.D.2). Then ψ ∈ M 1 (R).
Proof. We check each criteria in definition (C.D.4). By the exponential decay of
gj in the L1 -norm as j → ∞, the function ψ is clearly an L1 -function. Each gj is
supported in [−1, 1], and by the convolution theorem, the Fourier transform of gj
reads
2 −j
2
j/2 sin (2 πω)
.
gbj (ω) = 2j/2 χ[−2−(j+1) ,2−(j+1) b(ω)
=
2
]
(πω)2
To compute the L1 -norm of gbj (or equivalently the A(R)-norm of gj ), we utilize the
R
well-known identity for the sinc-integral, namely R sin(t)
t dt = π. After integration
by parts, we obtain
Z
Z
sin2 (2−j πω)
sin(21−j πω)
j/2
−j/2
∥gbj ∥1 = ∥gj ∥A(R) = 2
dω
=
2
dω = 2−j/2 .
(πω)2
πω
R
R
P
Thus, gj ∈ A(R), and the sum of the norms j ∥gj ∥A(R) also converges.
Proceeding, we show that Vψ is not subaveraging in M 1 (R):
Lemma C.D.5. There are functions ϕ ∈ M 1 (R) \ {0} such that for every fixed
0<R≤1
"Z
#−1
sup
f ∈M 1 (R)\{0}, z∈R2
|Vϕ f (z)| ·
|Vϕ f | dA(ξ)
= ∞.
BR (z)
In particular, the above statement holds for ϕ = ψ, where ψ is defined in (C.D.2).
109
Paper C. An FUP for the STFT and Gabor multipliers
Proof. Note that the ”local” norm behaviour of ψ, from (C.D.1) and (C.D.3), is
preserved in Vψ if we let the STFT act on the Dirac delta distribution δ ∈ S ′ (R),
where
Vψ δ(x, ω) = ψ(−x) without any ω-dependence.
Although, the Dirac delta itself does not belong to M 1 (R), we may approximate
it with functions from the Schwartz’ class. Namely, for any γ > 0 we consider
approximations {fγ }γ ⊆ S (R) of the form
Z
Z γ
fγ > 0,
fγ = 1 and
fγ > 1 − θ(γ) such that θ(γ) ↘ 0 as γ → 0.
R
−γ
For any bounded, continuous functions ϕ it is rather straightforward to show that
|⟨fγ , ϕ⟩| − |ϕ(0)| ≤ 2∥ϕ∥∞ θ(γ) + max |ϕ(t)| − |ϕ(0)| .
|t|≤γ
In our context, we consider ϕ = Mω Tx ψ so that
|Vψ fγ (x, ω)| − |ψ(−x)| ≤ 2∥ψ∥∞ θ(γ) + max |ψ(t − x)| − |ψ(−x)| .
|t|≤γ
Thus, for any j ∈ N0 and any ϵ > 0, there exists γ = γ(j, ϵ) > 0 such that
Vψ fγ(j,ϵ) (x, ω) − ψ(−x) ≤ ϵ∥gj ∥1 ∀ (x + 2j)2 ≤ R2 .
With this approximation, we consider the point evaluation of Vψ fj,ϵ at the point
zj := (−2j, 0) and compare it to the integral over the ball BR (zj ). Utilizing the
”local” norm properties of ψ, i.e., the norm properties of gj , we find the lower
bound
"Z
#−1
∥gj ∥∞
1−ϵ
|Vψ fγ(j,ϵ) (zj )| ·
|Vψ fγ(j,ϵ) (ξ)| dA(ξ)
≥
∥g
∥
2R
+ ϵπR2
j 1
BR (zj )
= 2j
1−ϵ
.
2R + ϵπR2
Since ϵ > 0 is arbitrary, the subaveraging constant C(R) of Vψ must be bounded
from below by 2j−1 /R. Furthermore, since j ∈ N0 is arbitrary, the constant C(R)
is, in fact, unbounded.
110
Paper D
Notes on Hardy’s uncertainty principle for the Wigner
distribution and Schrödinger evolutions
Helge Knutsen
Published in
Journal of Mathematical Analysis and Applications, 2023, volume 525,
issue 1.
Paper D
Notes on Hardy’s uncertainty
principle for the Wigner
distribution and Schrödinger
evolutions
Abstract
For Schrödinger equations with real quadratic Hamiltonians, it is known
that the Wigner distribution of the solution at a given time equals, up to a
linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy’s uncertainty principle for the joint time-frequency
representation, we present a general uniqueness result for such Schrödinger
equations, where the solution cannot have strong decay at two distinct times.
This approach gives new proofs to known, sharp Hardy type estimates for
the free Schrödinger equation, the harmonic oscillator and uniform magnetic
potentials, as well as new uniqueness results.
D.1
Introduction
Hardy’s uncertainty principle is originally formulated as a sharp decay estimate of
a function f and its Fourier transform fˆ. Normalizing the Fourier transform by
Z
fˆ(ξ) := (2π)−d
f (x)e−i⟨ξ,x⟩ dx,
(D.1.1)
Rd
it states that the decay of (f, fˆ) cannot exceed
2
2
|f (x)| = O (e−α|x| ) and |fˆ(ξ)| = O (e−β|ξ| ) for 4αβ = 1.
Later, at the beginning of this century in [30] and [15], Hardy’s uncertainty principle has taken a different interpretation, where the statement has been shown to
113
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
be equivalent to a sharp uniqueness result on the free Schrödinger equation
∂u
(x, t) = i∆u(x, t).
∂t
(D.1.2)
More precisely, at two distinct times t = 0 and t = T , the solution u of (D.1.2)
cannot exceed the decay conditions
2
2
|u(x, 0)| = O (e−α|x| ) and |u(x, T )| = O (e−β|x| ) for (4T )2 αβ = 1.
For an in-depth discussion on this dynamical interpretation and different proofs of
Hardy’s uncertainty principle, we refer to the recent survey-paper [38]. Starting
with [30], and in the sequel of papers [31], [32], [33], Escauriaza, Kenig, Ponce and
Vega have studied Schrödinger evolutions for more general Hamiltonians, which
include a bounded potential. Their scheme is based on establishing logarithmicconvexity properties of the solution of the Schrödinger equation, from which they
successfully derive uniqueness results similar to the free case. In the same spirit, in
[13] and more recently in [14], Cassano and Fanelli consider Schrödinger evolutions
of the harmonic oscillator and of systems with a magnetic potential, in addition
to some bounded perturbations of these. In particular, the magnetic potential is
given by a coordinate transformation A : Rd → Rd so that we consider the twisted
Laplacian ∆A := (∇ − iA(x))2 . The associated Schrödinger equation (with a
bounded perturbation V : Rd × R → C) then reads
∂u
(x, t) = i ∆A + V (x, t) u(x, t).
∂t
With the exception of some specific examples, and under some restraints on the
coordinate transform A, the uniqueness results for the magnetic potential are all
derived for dimension d ≥ 3. Note that, if we assume A to be a linear transformation (and disregard any perturbations), both the harmonic oscillator and the
magnetic potential represent systems with a quadratic Hamiltonian, which is the
focus of the present paper.
We also mention the recent breakthroughs by Kulikov, Oliveira and Ramos
in [66] on a conjecture posed by Vemuri [83] on the evolution of the Harmonic
oscillator system. Their result is based on similar changes of variables also present
in Cassano and Fanelli’s papers, which reduces the problem to a question for
the free particle case. Interestingly, the statements made in [66], [83] describes
evolution of the quantum system based solely on decay conditions on the initial
data f (x) := u(x, 0) and the Fourier transform fˆ(ξ), rather than decay conditions
at two distinct times.
Proceeding, for real quadratic Hamiltonians (without any perturbations) we
present an approach based on Hardy’s uncertainty principle for the Wigner distribution. Here we utilize that for such Schrödinger equations, the Wigner distribution of the solution equals, up to a linear coordinate transform, the Wigner
distribution of the initial condition. This description of the solution in terms of the
Wigner distribution first came to our attention through the works of Cordero and
Rodino, and later Cordero, Giacchi and Rodino, in their two-part series [21], [17]
114
D.2. Preliminaries
on the Wigner analysis of operators. From here it turns out to be remarkably simple to formulate a general uniqueness results for quadratic systems. Nonetheless,
considering specified quadratic Hamiltonians, it can still be demanding to produce
explicit Hardy type estimates.
The remainder of the text is organized as follows: In Section D.2, we cover necessary background theory. Namely, we introduce the Weyl quantization procedure
(Section D.2.1), and introduce relevant theory from symplectic mechanics (Section
D.2.2), which seems to be a natural perspective when considering real quadratic
quantum systems. We also briefly discuss Hardy’s uncertainty principle in relation to the Wigner distribution (Section D.2.3). Section D.3 contains the general
uniqueness result, where the subsequent and final Section D.4 is devoted to specific
examples of Schrödinger evolutions. In Section D.4, we reproduce known, sharp
Hardy type estimates for the free case, the harmonic oscillator and the uniform
magnetic potential. In addition, based on Williamson’s diagonalization theorem,
we present an explicit scheme for systems based on real, symmetric positive definite
matrices, with one final example for dimension d = 2.
D.2
D.2.1
Preliminaries
Weyl quantization and the Schrödinger equation
For the position and momentum observables x, p ∈ Rd , we associate the following
pseudodifferential operator
xj → x
bj = xj and pj → pbj = −iℏ
∂
.
∂xj
For their composition xj pk , we consider standard Weyl quantization
xj pk →
1
x
bj pbk + pbk x
bj .
2
Although we only require the quantization procedures outlined above, we menb with symbol
tion that in the general case the Weyl pseudodifferential operator H
function H = H(x, p) is given by
Z
i
x+y
−d
b
Hf (x) = (2πℏ)
H
, ξ e ℏ ⟨x−y,ξ⟩ f (y)dydξ.
(D.2.1)
2
R2d
In the context of the quantum mechanics, we consider symbol functions that represent the Hamiltonian of a physical system. The Schrödinger equation with Hamiltonian H is then given by
iℏ
∂u
b
(x, t) = Hu(x,
t),
∂t
(D.2.2)
where the solution u is a function (or distribution) of the variables x and t, subject to some initial condition, e.g., u(x, 0) = u0 (x). We shall focus on the case
115
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
P
ajk xj pk for real ajk ’s.
1
⟨M z, z⟩ for z = (x, p),
2
(D.2.3)
when the Hamiltonian is quadratic, namely, H(x, p) =
Alternatively, this can be expressed as
H(z) =
j,k
and M is some real-valued 2d × 2d symmetric matrix. On this form, symplectic
mechanics naturally enters the picture, and we reference [43] for a comprehensive
introduction.
D.2.2
Symplectic mechanics
We briefly cover some of the basic terminology and results from symplectic mechanics necessary to describe solutions of the Schrödinger equation (D.2.2) with
quadratic Hamiltonians (D.2.3). This is based on de Gosson’s book [43], specifically Chapter 1-3, 7 and 15.
Let J denote the standard symplectic matrix
0 I
J :=
,
−I 0
where 0 and I are the d × d zero and identity matrices, respectively. A useful
observation is that the inverse of J coincides with the transpose so that J −1 =
J T = −J. The symplectic (Lie) group Sp(2d, R) is a closed subgroup of the general
linear group GL(2d, R), that consists of all matrices S such that
S T JS = J.
Since the inverse S −1 is also symplectic, this latter condition turns out to be
equivalent to SJS T = J. Writing the matrix S on block form
A B
S=
,
C D
it is straightforward to verify that symplectic matrices are characterized by the
conditions
AT C, B T D are symmetric and AT D − C T B = I
⇐⇒
T
AB , CD
T
(D.2.4)
T
T
are symmetric and AD − BC = I.
From this characterization, we also deduce that the inverse of S is given by
T
D
−B T
−1
S =
.
(D.2.5)
−C T
AT
The standard symplectic form is denoted by σ(z, z ′ ) := ⟨Jz, z ′ ⟩. A basis {ej , fj }dj=1
of Rd × Rd is called a symplectic basis if
σ(ej , ek ) = 0 ∧ σ(fj , fk ) = 0 ∧ σ(ej , fk ) = −δjk ∀ j, k = 1, 2, . . . , d.
116
D.2. Preliminaries
One simple example is the canonical symplectic basis {(cj , 0), (0, cj )}dj=1 , where
{cj }j is the canonical basis for Rd .
An important family of symplectic matrices are the so-called ”free” symplectic
matrices. These are matrices S ∈ Sp(2d, R) which on block form satisfy
A B
S=
, det B ̸= 0,
(D.2.6)
C D
and every symplectic matrix can in fact be written as a product of two such
matrices. To each free matrix S on the form (D.2.6), we associate a quadratic
form W (x, x′ ), called the generating function of S, given by
W (x, x′ ) :=
1
1
⟨DB −1 x, x⟩ − ⟨B −1 x, x′ ⟩ + ⟨B −1 Ax′ , x′ ⟩.
2
2
Conversely, starting out with a quadratic form
W (x, x′ ) =
1
1
⟨P x, x⟩ − ⟨L−1 x, x′ ⟩ + ⟨Qx′ , x′ ⟩,
2
2
(D.2.7)
with real matrices such that P = P T and Q = QT , we can generate a corresponding
free symplectic matrix
LQ
L
SW =
.
(D.2.8)
P LQ − (L−1 )T P L
Based on the generating function W , we associate unitary operators on L2 (Rd ) to
the free matrix SW . Referencing the generating function in (D.2.7) and by scaling
according to the constant ℏ, these operators read
Z
p
−1 m
′
i
−d
b
2
SW,m u(x) := (2πiℏ)
| det L| i
e ℏ W (x,x ) u(x′ )dx′ ,
(D.2.9)
Rd
where m is an integer so that
mπ ≡ arg(det L)
mod 2π.
We refer to the operators SbW,m
√ as the quadratic Fourier transforms of SW , and by
√
fixing a branch of z so that i is well-defined, we obtain two operators associated
to S√
W that only differ by a sign. In particular, we choose the branch such that
arg( i) ≡ π4 mod 2π. Since the inverse exists
−1
SbW,m
= SbW ∗ ,m∗ for W ∗ (x, x′ ) = −W (x′ , x) and m∗ = m − d,
we can form a group from compositions of the quadratic Fourier transforms of the
free symplectic matrices. This group is known as the metaplectic group Mp(2d, R),
and it forms a double cover of the symplectic group Sp(2d, R) by extending the
map SbW,m → SW to a surjective group homomorphism
π Mp : Mp(2d, R) → Sp(2d, R), with ker(π Mp ) = {±I}.
117
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
To see how the symplectic and metaplectic group relates to the Schrödinger
equation with quadratic Hamiltonian H, we first recall the general Hamiltonian
equations determining the time evolution of a point z = (x, p), namely
∂xj
∂H
∂pj
∂H
=
and
=−
for j = 1, 2, . . . , d,
∂t
∂pj
∂t
∂xj
or more compactly
∂z
= J∇H with ∇ = (∂x1 , . . . , ∂xd , ∂p1 , . . . , ∂pd ).
∂t
When the Hamiltonian is on the form H(z) = 12 ⟨M z, z⟩, where M is a realsymmetric matrix, the Hamiltonian equations reduce to ∂z
∂t = JM z. Starting
at time t = 0 from the point z(0) = z0 , it is clear that z(t) = StH (z0 ) with
StH := exp(tJM ) solves the initial value problem.
We now consider the Lie algebra of the symplectic group, also known as the
symplectic algebra, denoted by sp(2d, R), which consists of all matrices X such
that
XJ + JX T = 0.
By inspection, we find that the matrix JM (and also tJM ) belongs to sp(2d, R).
Conversely, for any X ∈ sp(2d, R), we have that JX is symmetric. Hence, any
quadratic Hamiltonian H can be expressed as
1
H(z) = − ⟨JXz, z⟩ for some X ∈ sp(2d, R).
2
(D.2.10)
Since the exponential maps the symplectic algebra into the symplectic group, we
must have that the operator StH = exp(tX) is symplectic for any fixed t, and in
turn StH t≥0 is a one-parameter subgroup of Sp(2d, R).
We also consider the Lie algebra of the metaplectic group, or simply the metaplectic algebra, denoted by mp(2d, R). An essential observation is that the metaplectic algebra mp(2d, R) is isomorphic to sp(2d, R), which follows from the fact
that Mp(2d, R) is a covering group of Sp(2d, R). In particular, there is an explicit
isomorphism F : sp(2d, R) → mp(2d, R) so that the following diagram commutes:
mp(2d, R)
F −1
exp
Mp(2d, R)
sp(2d, R)
exp
π
Mp
(D.2.11)
Sp(2d, R)
What follows are the main results of the section. For details and proofs, we refer
to Chapter 15 in [43] (especially Theorem 352 and Corollary 355).
Theorem D.2.1. Let H be the quadratic Hamiltonian of the form
1
H(z) = − ⟨JXz, z⟩ for some X ∈ sp(2d, R).
2
118
D.2. Preliminaries
b where
The linear mapping F , which to X associates the operator F (X) := − ℏi H
Weyl
b ←
H
−−→ H, then satisfies
[F (X), F (X ′ )] = F ([X, X ′ ]) ∀ X, X ′ ∈ sp(2d, R),
and F forms an isomorphism sp(2d, R) → mp(2d, R) so that diagram (D.2.11)
commutes.
Based on the isomorphism F and since diagram (D.2.11) commutes, we can
describe the solution of Schrödinger equation (D.2.2) with quadratic Hamiltonians
as a lift of the flow t 7→ StH in Sp(2d, R) into the unique path t 7→ SbtH in Mp(2d, R)
so that Sb0H = I.
Corollary D.2.1. Let the Hamiltonian H be quadratic, and let t 7→ SbtH be the
lift to Mp(2d, R) of the flow t 7→ StH . Then for any u0 ∈ S (Rd ), the function
u(x, t) = SbtH u0 (x) is a solution of the initial value problem
iℏ
∂u
b
(x, t) = Hu(x,
t), u(x, 0) = u0 (x).
∂t
In particular, whenever exp(tJM ) is a free symplectic matrix, we can express
the solution of (D.2.2) as a quadratic Fourier transform (D.2.9) of the initial condition.
D.2.3
Hardy’s uncertainty principle and the Wigner
distribution
For a function f : Rd → C, we normalize the Fourier transform according to
(D.1.1). Among the many uncertainty principles, Hardy’s uncertainty principle is
a precise statement regarding the largest possible decay of the pair (f, fˆ). The
original 1933-paper [56] by Hardy covers the 1-dimensional case, which has later
been extended to higher dimensions (see [40]).
Theorem D.2.2. (Hardy’s Uncertainty Principle) Suppose f ∈ L2 (Rd ) satisfies
the decay conditions
2
|f (x)| ≤ Ke−α|x|
2
and |fˆ(ξ)| ≤ Ke−β|ξ|
for some constants α, β, K > 0.
(i) If 4αβ > 1, then f ≡ 0.
2
(ii) If 4αβ = 1, then f = ce−α|x| for some c ∈ C.
In the joint time-frequency representation, there are analogous statements to
Hardy’s uncertainty principle. While these may originally have been formulated
in terms of the Short-Time Fourier Transform or the cross-Ambiguity function,
we shall present them equivalently in terms of the cross-Wigner distribution. We
normalize the cross-Wigner distribution using ℏ to match our Weyl quantization
119
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
procedure in (D.2.1) for a physical system (otherwise we can think of ℏ = 1).
Namely, for a pair (f, g) ∈ S × S ′ (Rd ) the cross-Wigner distribution is given by
W ℏ (f, g)(x, ξ) := (2πℏ)−d
Z
i
y
y e− ℏ ⟨ξ,y⟩ f x +
dy.
g x−
2
2
Rd
In the case when f = g, we write W ℏ (f, f ) = W ℏ f and refer to it simply as
the Wigner distribution or Wigner transform of f . The first analog of Hardy’s
uncertainty principle in the joint representation was discovered by Gröchenig and
Zimmermann in [52] (see Theorem 1.2 and Corollary 3.3).
Theorem D.2.3. Suppose (f, g) ∈ S × S ′ (Rd ) such that
2
|W ℏ (f, g)(x, ξ)| ≤ Ke−(α|x|
+β|ξ|2 )
for some constants α, β, K > 0.
(i) If αβ > ℏ−2 , then W ℏ (f, g) ≡ 0 so f ≡ 0 or g ≡ 0.
(ii) If αβ = ℏ−2 and W ℏ (f, g) ̸≡ 0, then both f and g are multiples of a time2
α
frequency shift of the Gaussian e− 2 |x| , that is, f and g are multiples of
2
α
ei⟨ξ0 ,x⟩ e− 2 |x−x0 | for some constants ξ0 , x0 ∈ Rd .
Later in [11], several estimates on the largest possible decay of the Ambiguity
function (or equivalently the Wigner distribution) have been derived. One of these
results separates the decay conditions in the x- and ξ-direction (see Corollary 6.5.).
Theorem D.2.4. Suppose f, g ∈ L2 (Rd ) such that
2
Z
W ℏ (f, g)(x, ξ) e2π|xj |
Z
dxdξ < ∞ and
(1 + |xj |)M
R2d
W ℏ (f, g)(x, ξ) e(2π)
2
−1 −2
ℏ
|ξj |2
(1 + |ξj |)N
R2d
2
dxdξ < ∞
for some j = 1, . . . , d. If min{M, N } ≤ 1, then W ℏ (f, g) ≡ 0 so f ≡ 0 or g ≡ 0.
From the above result, we easily deduce sufficient decay conditions for the
Wigner distribution similar to that of Theorem D.2.3, but with the x- and ξdirection separated.
Corollary D.2.2. Suppose f, g ∈ L2 (Rd ) such that
|W ℏ (f, g)(x, ξ)| ≤ Ke−
P
j
αj |xj |2
and |W ℏ (f, g)(x, ξ)| ≤ Ke−
P
j
βj |ξj |2
for some constants αj , βj , K > 0. If for some j = 1, . . . , d the product αj βj > ℏ−2 ,
then W ℏ (f, g) ≡ 0 so f ≡ 0 or g ≡ 0.
120
D.3. Hardy’s uncertainty principle for quadratic Hamiltonians
What is remarkable about the latest statement is that, although we require
decay in every direction, it suffices to consider the largest combined decay for any
given pair (xj , ξj ) to conclude that the Wigner distribution is zero. Such conditions
have also been derived for the separate representation in [45], [46] based on the
symplectic capacity of the ellipsoid associated to the exponents.1 In the same vein,
in [44], a similar result to Corollary D.2.2 is obtained for the Wigner distribution.
We shall utilize the uncertainty principle stated in Corollary D.2.2 when we derive
uniqueness results for the Schrödinger equation with quadratic Hamiltonians.
D.3
Hardy’s uncertainty principle for quadratic
Hamiltonians
We shall prove the following Hardy type estimate:
Theorem D.3.1. Let u(·, t) ∈ S (Rd ) be the solution of the Schrödinger equation
(D.2.2) with quadratic Hamiltonian H(z) = − 21 ⟨JXz, z⟩ for some X ∈ sp(2d, R).
Suppose at time t = 0 and time t = T , the solution u satisfies the decay conditions
2
|u(x, 0)| ≤ Ke−α|x|
2
and |u(x, T )| ≤ Ke−β|x|
for some constants α, β, K > 0. If the exponential
· B(T )
exp(T X) =
is free symplectic and
·
·
(2ℏ)2 ∥B(T )∥2op αβ > 1,
then u ≡ 0.
The proof is based on Corollary D.2.1, where the solution of the Schrödinger
equation can be written as a metaplectic transform of the initial condition. On
this form, the proof is divided into two lemmas. The first lemma, Lemma D.3.2,
is well known and referred to as the covariance property of the Wigner distribution, where the Wigner distribution composed with a metaplectic transformation
corresponds to the Wigner distribution with an associated symplectic coordinate
transformation. This shows that the Wigner distribution of the solution equals,
up to a symplectic coordinate transform, the Wigner distribution of the initial
condition. In the second lemma, Lemma D.3.3, we combine this fact with Hardy’s
uncertainty principle for the Wigner distribution.
Lemma D.3.2. (Covariance property; see, e.g., Corollary 217 in [43]) Let Sb ∈
b denote the projection of Sb on Sp(2d, R). Then for
Mp(2d, R), and let S = π Mp (S)
d
b satisfies
any u ∈ S (R ), the Wigner distribution of Su
b
W ℏ (Su)(x,
ξ) = W ℏ u(S −1 (x, ξ)).
(D.3.1)
1 In the separate representation, there is a much more general result stated for tempered
distributions, see Corollary 1.6.9. in [23]. Notably, if we restrict to functions in L2 (Rd ), it
suffices to have large enough decay for one pair (xj , ξj ) with no decay condition in the other
directions to conclude that the function is zero.
121
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
Observe that by formula (D.2.5), the inverse of the free symplectic
matrix
T
exp(X) = ( ·· B· ) is also free symplectic such that exp(X)−1 = ·· −B
. Hence,
·
Theorem D.3.1 follows once we prove the next lemma.
Lemma D.3.3. Let S −1 be a free symplectic matrix on the form
A B
−1
S =
, det B ̸= 0.
C D
Suppose that for two functions u0 , u1 ∈ S (Rd ) their Wigner distributions satisfy
the identity
W ℏ u1 (x, ξ) = W ℏ u0 S −1 (x, ξ) .
Suppose further that the functions satisfy the decay conditions
2
|u0 (x)| ≤ Ke−α|x|
and |u1 (x)| ≤ Ke−β|x|
2
for α, β, K > 0.
If (2ℏ)2 αβ · ∥B∥2op > 1, then u0 , u1 ≡ 0.
Proof. By the decay conditions on u0 and u1 , we easily deduce that the associated
Wigner distributions are bounded by
2
(i) |W ℏ u0 (x, ξ)| ≤ κe−2α|x| and
2
(ii) |W ℏ u1 (x, ξ)| ≤ κe−2β|x| for some constant κ > 0.
Similarly to the proof of Hardy’s uncertainty principle for the free Schrödinger
equation in Section 2.2 in [38], wherein the initial condition u(x, 0) is written as
2
u(x, 0) = eia|x| f (x), we express
i
u0 (x) = e ℏ ⟨M x,x⟩ f (x) for some f ∈ S (Rd ),
and M is some real d × d matrix to be decided. Evidently, |u0 (x)| = |f (x)|, and
we shall therefore prove the Hardy type estimate for f . On this form, the Wigner
distribution of u0 reads
W ℏ u0 (x, ξ) = W ℏ f x, ξ − (M + M T )x .
Since the right-hand side of (i) is independent of ξ, we may set ξ := ω +(M +M T )x
and maintain the same decay condition for W ℏ f as for W ℏ u0 , namely
2
|W ℏ f (x, ω)| ≤ κe−2α|x| .
(D.3.2)
Similarly, we express the Wigner distribution of u1 in terms of f , and utilizing the
identity W ℏ u1 (x, ξ) = W ℏ u0 (S −1 (x, ξ)), we find that
W ℏ u1 (x, ξ) = W ℏ f Z(x, ξ) ,
where
Z=
122
A
C − (M + M T )A
B
.
D − (M + M T )B
D.4. Examples of Schrödinger evolutions
We now choose the matrix M such that D − (M + M T )B = 0, which is possible
since B is invertible. In particular, we have that
M + M T = DB −1 ,
which shows that the matrix DB −1 is symmetric, and in fact, we could have
chosen M to be symmetric. With this choice of M , the lower left block of Z is
given by C − DB −1 A. By the characterization of symplectic matrices (D.2.4) and
since S −1 is free symplectic, we may express C in terms of A, B and D, namely
C = (DB −1 )T A − (B −1 )T . Thus, the lower left block simplifies to −(B −1 )T , and
the matrix Z in turn simplifies to
A
B
Z=
,
−(B −1 )T 0
so that
W ℏ u1 (x, ξ) = W ℏ f Ax + Bξ, −(B −1 )T x .
Again since the right-hand side of (ii) is independent of ξ, define
ξ := B −1 y − B −1 Ax,
so the decay condition reads
2
|W ℏ f (y, −(B −1 )T x)| ≤ κe−2β|x| ,
or equivalently
|W ℏ f (y, ω)| ≤ κe−2β|B
T
ω|2
.
(D.3.3)
By combining the two decay conditions (D.3.2) and (D.3.3), the statement now
follows from Corollary D.2.2.
D.4
Examples of Schrödinger evolutions
In this section we provide explicit examples of quadratic Hamiltonians and what
the associated Hardy type estimate of Theorem D.3.1 look like.
D.4.1
Free particle, harmonic oscillator and magnetic
potential
To begin with, we consider some cases where there are known Hardy type estimates. We consider the notorious free particle case and a generalized harmonic
oscillator, which has been studied by Cassano and Fanelli in [14] for the special
case where all angular frequencies are equal. Finally, we consider the uniform
magnetic potential, also covered in [14].
123
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
Example D.4.1. (Free Schrödinger equation) Consider a system without any
external potential, that is, consider the Hamiltonian H of the form
H(z) =
1
1
|p|2 =
p21 + · · · + p2d , where z = (x, p).
2m
2m
This Hamiltonian corresponds to the so-called free Schrödinger equation
iℏ
∂u
ℏ2
(x, t) = −
∆u(x, t).
∂t
2m
(D.4.1)
Expressing the Hamiltonian instead as the inner product H(z) = 21 ⟨M z, z⟩, we
have that
1
0 0
0 m
I
M=
,
and
consequently
X
:=
JM
=
∈ sp(2d, R).
1
I
0 0
0 m
Since X 2 = 0, the exponential exp(tX) reduces to
t
I m
I
exp(tX) = I + tX =
,
0 I
which is free symplectic for all t > 0. Hence, by Theorem D.3.1, we obtain the
following statement for the free Schrödinger equation:
Corollary D.4.1. Let u(·, t) ∈ S (Rd ) be the solution of the free Schrödinger
equation (D.4.1). Suppose at time t = 0 and time t = T , the solution u satisfies
the decay conditions
2
|u(x, 0)| ≤ Ke−α|x|
for some constants α, β, K > 0. If αβ
and |u(x, T )| ≤ Ke−β|x|
2ℏT 2
m
2
> 1, then u ≡ 0.
Remark. By comparison with the already established Hardy’s uncertainty principle for the free case, with m = 12 and ℏ = 1 (see Theorem 3 in [38]), we have
rediscovered the same condition on the exponents α, β. Furthermore, as the literature shows, our Hardy type estimate is, in fact, sharp.
Example D.4.2. (Harmonic Oscillator) Consider now the Hamiltonian given by
H(z) =
m 2 2
1
p21 + · · · + p2d +
ω1 x1 + · · · + ωd2 x2d , where z = (x, p).
2m
2
This is known as the harmonic oscillator, and the associated Schrödinger equation
reads
∂u
ℏ2
m 2 2
2 2
iℏ (x, t) = −
∆+
ω1 x1 + · · · + ωd xd u(x, t).
(D.4.2)
∂t
2m
2
Define for simplicity the diagonal matrix Ω := diag(ωj ). Thus, on the inner
product form H(z) = 12 ⟨M z, z⟩, the matrix M can be written
1
mΩ2
0
0
mI
M=
,
and
also
X
:=
JM
=
∈ sp(2d, R).
1
0
−mΩ2
0
mI
124
D.4. Examples of Schrödinger evolutions
P∞ k
For the power series exp(tX) = k=0 tk! X k , we distinguish between the matrices
with even and odd exponents, which, by induction, are given by
2k
1 2k
0
Ω
0
2k
k Ω
2k+1
k
m
X = (−1)
and X
= (−1)
.
0
Ω2k
−mΩ2(k+1)
Thus, the summation reads
∞
X
(−1)k t2k Ω2k
exp(tX) =
0
(2k)!
k=0
0
Ω2k
+
∞
X
(−1)k t2k+1
k=0
(2k + 1)!
1 2k
mΩ
0
−mΩ2(k+1)
0
.
Since Ω is diagonal, we may move
the summation
inside the matrix, and we find
A(t) B(t)
that each block in exp(tX) = C(t)
is
rather
easy
to compute. In particular,
D(t)
for the B(t)-block, we have that
∞
B(t) =
1 X (−1)k t2k+1 2k
Ω
m
(2k + 1)!
k=0
1
= diag
m
∞
1 X (−1)k (ωj t)2k+1
ωj
(2k + 1)!
k=0
!
1
= diag
m
sin(ωj t)
ωj
.
Similarly, for each of the other blocks, we recognize the Taylor series for the sine
and cosine function, so that
A(t) = D(t) = diag cos(ωj t) and C(t) = −m diag ωj sin(ωj t) .
Nonetheless, based solely on the B(t)-block and Theorem D.3.1, we obtain the
following Hardy type estimate for the harmonic oscillator:
Corollary D.4.2. Let u(·, t) ∈ S (Rd ) be the solution of the Schrödinger equation
(D.4.2) corresponding to the harmonic oscillator. Suppose at time t = 0 and time
t = T , the solution u satisfies the decay conditions
2
|u(x, 0)| ≤ Ke−α|x|
2
and |u(x, T )| ≤ Ke−β|x|
for some constants α, β, K > 0. If sin(ωj T ) ̸= 0 for all j = 1, 2, . . . , d and
αβ
2ℏ
m
2
max
j
sin(ωj T )
ωj
2
> 1, then u ≡ 0.
Remark. Notice that in Theorem D.3.1, we only require the matrix M in H(z) =
1
2 ⟨M z, z⟩ to be real-valued, symmetric. Thus, we may also consider the case when
ωj2 < 0, meaning we would consider an angular frequency ωj that is purely imaginary. In this case we would simply replace | sin(ωj T )| with | sinh(|ωj |T )| in Corollary D.4.2.
Remark. In contrast to the free Schrödinger equation, for the harmonic oscillator
there are times T > 0 where our procedure yields no Hardy type estimate. This is a
somewhat similar phenomenon to that of Theorem 1.1 in [66], where the evolution
125
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
result for the harmonic oscillator also breaks down for an ”exceptional” set of time
points. In our case these are exactly the time points when exp(T X) is no longer
free symplectic, i.e., the time points when sin(ωj T ) = 0. If ωj = ω > 0 for all
j, the solution is periodic in time, and these time points represent periods or half
periods of the solution, for which we do not expect any Hardy type estimate to
be present, as we are essentially trying to extract information from a single time
point. However, if ωj − ωk ∈
/ Q for some pair (ωj , ωk ), the solution is not even
periodic in time, and we do not have such a nice interpretation of why our Hardy
estimate breaks down.
Remark. As previously mentioned, Hardy’s uncertainty principle for the harmonic oscillator has also been studied by Cassano and Fanelli in [14], and with
m = 21 , ℏ = 1 and ωj2 = ω 2 > 0 for j = 1, . . . , d they present a sharp condition on
the exponents α, β (see Theorem 1.3 and Theorem 1.9). By comparison with the
above corollary, we again find that the condition coincides.
Example D.4.3. (Uniform magnetic potential) Consider the uniform magnetic
potential
A : Rd → Rd , x 7→ mωBx,
where B is some real-valued d × d matrix such that B T = −B and B T B = I. For
d = 2, the potential is of the form A(x1 , x2 ) = ±mω(−x2 , x1 ). The associated
Hamiltonian is given by
1
2
p + A(x)
2m
mω 2 2
1
=
|p|2 + ω ⟨Bx, p⟩ − ⟨Bp, x⟩ +
|x| , where z = (x, p),
2m
2
and Schrödinger equation reads
H(z) =
iℏ
2
∂u
1
(x, t) = −
∇ − iA(x) u(x, t)
∂t
2m
(D.4.3)
Expressing the Hamiltonian as the inner product H(z) = 21 ⟨M z, z⟩, it follows that
1
mω 2 I ωB
−ωB
mI
and
X
:=
JM
=
∈ sp(4, R).
M=
1
−ωB m
I
−mω 2 I −ωB
Since B 2 = −I, we easily deduce that
X 2k+1 = (−1)k (2ω)2k X and X 2k = (−1)k−1 (2ω)2(k−1) X 2 for k > 0.
Thus, we may follow a similar procedure to the harmonic oscillator case.
Corollary D.4.3. Let u(·, t) ∈ S (Rd ) be the solution of the Schrödinger equation
(D.4.3) corresponding to the uniform magnetic potential. Suppose at time t = 0
and time t = T , the solution u satisfies the decay conditions
2
|u(x, 0)| ≤ Ke−α|x|
for some constants α, β, K > 0. If sin(ωT ) ̸= 0 and αβ
u ≡ 0.
126
2
and |u(x, T )| ≤ Ke−β|x|
2
2ℏ 2 sin(ωT )
m
ω
> 1, then
D.4. Examples of Schrödinger evolutions
Remark. Again we compare with the results in [14] by Cassano and Fanelli for
the case m = 12 , ℏ = 1. From their Theorem 1.11, we have the same Hardy type
estimate, which is verified to be sharp in 2-dimensions by their Theorem 1.4.
D.4.2
Systems based on positive definite matrices
From the previous examples, it should be evident that providing an explicit Hardy
type estimate based on Theorem D.3.1 for a specific Hamiltonian H(z) = 12 ⟨M z, z⟩
is contingent upon our ability to compute the associated exponential, exp(tJM ).
For the free and harmonic oscillator case, this is rather simple since M , in both
cases, is a diagonal matrix. In general, this might be quite a challenging computation. However, for the family of positive definite matrices, i.e., matrices M such
that ⟨M z, z⟩ > 0 ∀ z ̸= 0, we may apply Williamson’s diagonalization theorem
to simplify our computations.
Theorem D.4.1. (Williamson’s diagonalization theorem; see, e.g., Theorem 93 in
[43]) Let M be a positive definite symmetric real 2d × 2d-matrix. The eigenvalues
of JM are all of the form ±iλj for λj > 0, and the associated eigenvectors can be
written (ej ± ifj ) so that {ej , fj }dj=1 forms a symplectic basis. The matrix
S := e1 | . . . |ed
f1 | . . . |fd ∈ Sp(2d, R)
(D.4.4)
then diagonalizes M such that
ST M S =
Λ 0
0 Λ
for Λ = diag(λj ).
(D.4.5)
From here, we obtain a closed form of the exponential.
Lemma D.4.2. Let M be a positive definite symmetric real 2d × 2d matrix, and
define the matrix X := JM . The eigenvalues of X are on the form ±iλj for
λj > 0, and the associated eigenvectors can be written (ej ± ifj ) so that {ej , fj }dj=1
forms a symplectic basis. Then for the matrix
S := e1 | . . . |ed f1 | . . . |fd ∈ Sp(2d, R),
(D.4.6)
we have the following matrix decomposition of the exponential
Θ −Ω
−1 T
exp(tX) = J(S )
S −1
Ω Θ
(D.4.7)
with Θ := diag(sin(λj t)), Ω := diag(cos(λj t)) and parameter t ∈ R.
Proof. By the diagonalization in Theorem D.4.1, we can write
Λ 0
X = J(S −1 )T
S −1 for Λ = diag(λj ).
0 Λ
127
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
By definition of symplectic matrices, S −1 J(S −1 )T = J, it follows that
X j = J(S −1 )T
Λ 0
0 Λ
j−1
Λ 0
J
S −1 for j = 1, 2, . . .
0 Λ
In turn, the power series of exp(tX) reads
exp(tX) =
∞ j j
X
t X
j=0
j!


∞ j
X
t
j−1
= I + J(S −1 )T 
( Λ0 Λ0 ) [J( Λ0 Λ0 )]  S −1 .
j!
j=1
We now compute the sum within the square brackets [. . . ], and similarly to the
harmonic oscillator, we distinguish between even and odd exponents, where
2k
2k
2k+1
0
Λ2k+1
[J( Λ0 Λ0 )] = (−1)k Λ0 Λ02k and [J( Λ0 Λ0 )]
= (−1)k −Λ2k+1
.
Thus, the sum can be expressed
∞ j
X
t
j=1
=
j!
j−1
( Λ0 Λ0 ) [J( Λ0 Λ0 )]
∞
X
(−1)k t2k+1 k=0
(2k + 1)!
Λ2k+1
0
0
Λ2k+1
−
∞
X
(−1)k t2k k=1
(2k)!
0
Λ2k
−Λ2k 0
.
By pulling the summation inside the matrices, we immediately recognize the Taylor
series of the sine and cosine function. In particular, with the same notation as in
(D.4.7), we find that
∞ j
X
t
j=1
j!
j−1
( Λ0 Λ0 ) [J( Λ0 Λ0 )]
=
Θ
0
0
0
+J −
Θ
−Ω
Ω
.
0
Again since S −1 is symplectic, we have that J (S −1 )T JS −1 = J 2 = −I, and
result (D.4.7) follows.
The purpose of the fourth and final example is twofold: Firstly, it illustrates
how we may apply Lemma D.4.2 to produce explicit Hardy type estimates based
on Theorem D.3.1. Secondly, the example does not seem to be covered by previous
literature.
Example D.4.4. (2D–harmonic oscillator with a cross-term) Consider the Hamiltonian
H(z) =
1
mω 2 2
p21 + p22 + θp1 x2 +
x1 + x22 where z = (x, p) ∈ R2 × R2 ,
2m
2
which corresponds to the Schrödinger equation
∂u
ℏ2
mω 2 2
∂
iℏ (x, t) = −
∆ − iℏ θx2
+
x1 + x22 u(x, t).
∂t
2m
∂x1
2
128
(D.4.8)
D.4. Examples of Schrödinger evolutions
This system is similar to the harmonic oscillator but with an added cross-term
+θp1 x2 in the Hamiltonian. With this additional term, the symmetric matrix M
in H(z) = 12 ⟨M z, z⟩ is no longer a diagonal matrix, rather, we have that


mω 2
0
0 0
 0
mω 2 θ 0 
,
M =
1
 0
θ
0
m
1
0
0
0 m
in addition to

0
 0

X := JM = 
−mω 2
0
θ
0
0
−mω 2
1
m
0
0
−θ
0

1
m
0
0
∈ sp(4, R).
Proceeding, we assume θ ∈ R such that ω > |θ|, making M a real positive definite
matrix, to which we may apply Lemma D.4.2. Consider first the following set of
vectors in R4 :
 1 


0
− mω
1  1 
12 
mp
p
0 
 , f1 := m ω(ω + θ)−1 2 − m  ,
ω(ω + θ) 
e1 :=
 0 
 −ω 
2
2
1
0
  (D.4.9)
 1 
0
−
mp
mp
1
12  mω 
−1 2  1 
0

 m .
e2 :=
ω(ω − θ) 
−ω 
 0  , f2 := 2 ω(ω − θ)
2
0
−1
It is straightforward to verify that (ej ± ifj ) constitute eigenvectors of X such that
X(ej ± ifj ) = ±iλj (ej ± ifj ),
p
p
where λ1 = ω(ω + θ) and λ2 = ω(ω − θ).
(D.4.10)
By the normalization of the vectors, we also find that {ej , fj }2j=1 forms a
symplectic basis for R2 × R2 . Thus, we may use
S := e1 |e2 |f1 |f2 ∈ Sp(4, R)
(D.4.11)
to decompose the exponential, exp(tX), according to Lemma D.4.2 such that
Θ −Ω
exp(tX) = J(S −1 )T
S −1
Ω Θ
with Θ := diag(sin(λj t)) and Ω := diag(cos(λj t)).
A B ) and utilizing expression (D.2.5) for the inverse
Writing S on block form S = ( C
D
−1
S , it follows that
· B(t)
B(t) = BΘB T + BΩAT − AΩB T + AΘAT , where exp(tX) =
.
·
·
129
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
By combining (D.4.9)–(D.4.11), the two upper blocks read
!
q
q
λ1
λ2
0
2
2
2mω
2mω
A=−
and B =
√ 1
−
0
0
2mλ1
0
√ 1
2mλ2
.
Subsequently, we obtain an explicit expression for the upper right block of the
exponential, exp(tX), namely
λ1
1
sin(λ1 t) + λω2 sin(λ2 t)
cos(λ2 t) − cos(λ1 t)
ω
B(t) =
, (D.4.12)
ω
ω
cos(λ1 t) − cos(λ2 t)
2mω
λ1 sin(λ1 t) + λ2 sin(λ2 t)
With this latest result, we are ready to compute for the Schrödinger equation
(D.4.8) an explicit Hardy type estimate based on Theorem D.3.1.
Corollary D.4.4. Let u(·, t) ∈ S (Rd ) be the solution of the Schrödinger equation
(D.4.8), with θ ∈ R such that ω > |θ|. Suppose at time t = 0 and time t = T , the
solution u satisfies the decay conditions
2
|u(x, 0)| ≤ Ke−α|x|
and |u(x, T )| ≤ Ke−β|x|
2
for some constants α, β, K > 0. Define the quantity
2(2mω)2 Γ± (T ) :=
ω 4 + λ41
ω 4 + λ42
2
sin
(λ
T
)
+
sin2 (λ2 T )
1
ω 2 λ21
ω 2 λ22
2
ω 4 + (λ1 λ2 )2
sin(λ1 T ) sin(λ2 T ) + 2 cos(λ1 T ) − cos(λ2 T )
2
(ωλ1 λ2 )
ω 2 − λ21
ω 2 − λ22
(D.4.13)
±
sin(λ1 T ) +
sin(λ2 T ) ·
ωλ1
ωλ2
!1
2
2
2 2
ω + λ21
ω 2 + λ22
·
sin(λ1 T ) +
sin(λ2 T ) + 4 cos(λ1 T ) − cos(λ2 T )
,
ωλ1
ωλ2
p
p
with λ1 = ω(ω + θ) and λ2 = ω(ω − θ). If Γ± (T ) ̸= 0 and αβ(2ℏ)2 Γ+ (T ) > 1,
then u ≡ 0.
+2
Proof. It remains to show that the eigenvalues of B T B(T ) coincide with Γ− (T ) ≤
Γ+ (T ) so that Γ+ (T ) = ∥B(T )∥2op . From (D.4.12), we see that the matrix B(T )
a b
is of the form B(T ) = −b
c , and in turn
2
a + b2 (a − c)b
T
B B(T ) =
.
(a − c)b b2 + c2
The two eigenvalues δ± of B T B(T ) are then given by
i
p
1h 2
a + 2b2 + c2 ± |a − c| (a + c)2 + 4b2 ,
δ± =
2
and the result follows once we replace a, b and c with the expressions in (D.4.12).
Remark. Although the quantity Γ± (T ) in (D.4.13) looks rather daunting, it is
easy to see that the expression reduces to the harmonic oscillator case of Corollary
D.4.2 when θ → 0.
130
D.4. Examples of Schrödinger evolutions
Acknowledgements
The research of the author was supported by Grant 275113 of the Research Council
of Norway. The research was in part conducted while visiting the Department of
Mathematics at Stanford University. The visit was partially funded by NSF, Grant
DMS-1956294, and the author has also received support from the project Pure
Mathematics in Norway – Ren matematikk i Norge funded by the Trond Mohn
Foundation, Grant TMS2021TMT03. The author would like to extend thanks to
prof. Eugenia Malinnikova and prof. Franz Luef for the many insightful discussions
and feedback on early drafts of manuscript. Finally, the author is grateful to the
anonymous referee who suggested to include uniform magnetic potentials.
131
Paper D. Hardy’s UP for the Wigner distribution and Schrödinger evolutions
132
Appendix
D.A
Williamson’s diagonalization theorem
We recap Williamson’s diagonalization theorem with a particular focus on the
constructive nature of the proof.
Theorem D.A.1. (Williamson’s diagonalization theorem) Let M be a positive
definite symmetric real 2d × 2d-matrix. The eigenvalues of JM are all on the form
±λj for λj > 0, and the associated eigenvectors can be written (ej ± ifj ) so that
{ej , fj }dj=1 forms a symplectic basis. The matrix
S := e1 | . . . |ed
f1 | . . . |fd ∈ Sp(2d, R)
(D.A.1)
then diagonalizes M such that
Λ 0
S MS =
for Λ = diag(λj ).
0 Λ
T
(D.A.2)
Proof. Define the inner product ⟨·, ·⟩M on C2d by ⟨z, z ′ ⟩M := ⟨M z, z ′ ⟩. Since
both ⟨·, ·⟩M and the standard symplectic form σ(·, ·) are non-degenerate, there
exists a unique invertible matrix K such that ⟨z, Kz ′ ⟩M = σ(z, z ′ ). Evidently,
the matrix K satisfies K T M = J = −M K, and the ⟨·, ·⟩M -transpose is given by
K M = −M −1 K T M = −K. Considering the positive self-adjoint operator K M K,
there exists an ⟨·, ·⟩M -orthonormal real eigenbasis {vj }j so that K M Kvj = µ2j vj
for some µj > 0. Since K M K = −K 2 and by the definition ⟨z, Kz ′ ⟩M = σ(z, z ′ ), it
follows that vj and Kvj form ⟨·, ·⟩M -orthogonal eigenvectors corresponding to the
same eigenvalue µ2j . Now, choose eigenvectors v1 , . . . , vd such that Kv1 , . . . , Kvd ∈
/
−1
d
span{v1 , . . . , vd }. Then, the set {vj , µj Kvj }j=1 makes up an ⟨·, ·⟩M -orthonormal
eigenbasis for K M K. Making the following normalization
1
1
ej := √ vj and fj := √ µ−1
Kvj ,
µj
µj j
we find that for all j, k = 1, 2, . . . , d
σ(ej , ek ) = 0 ∧ σ(fj , fk ) = 0 ∧ σ(ej , fk ) = −δjk ,
meaning the set {ej , fj }dj=1 forms a symplectic basis. Let S ∈ Sp(2d, R) be the
matrix that maps the canonical symplectic basis to the new basis {ej , fj }j . The
⟨·, ·⟩M -orthogonality of {ej , fj }j then verifies the diagonalization (D.A.2) with Λ =
diag(µ−1
j ). We can verify directly that K(fj ± iej ) = ±iµj (fj ± iej ), which is
−1
equivalent to K −1 (ej ±ifj ) = ±iµ−1
= JM , we are done.
j (ej ±ifj ). Since K
133
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