Frames and operators:
Basic properties and open problems
Ole Christensen
Department of Mathematics
Technical University of Denmark
Denmark
July 16, 2012
(DTU Mathematics)
Talk, Concentration Week, Texas A & M
July 16, 2012
1 / 85
Abstract The lectures will begin with an introduction to frames in general
Hilbert spaces. This will be followed by a discussion of frames with a special
structure, in particular, Gabor frames and wavelet frames in L2 (R)..
The lectures will highlight the connections to operator theory, and also
present some open problems related to frames and operators.
(DTU Mathematics)
Talk, Concentration Week, Texas A & M
July 16, 2012
2 / 85
Charlotte, NC, July 1996
(DTU Mathematics)
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July 16, 2012
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Plan for the talks
• Part I: Frames in general Hilbert spaces
• Part II: Gabor frames and wavelet frames in L2 (R)
• Part III: Research topics related to frames and operator theory
(DTU Mathematics)
Talk, Concentration Week, Texas A & M
July 16, 2012
4 / 85
Plan for the talks
• Part I: Frames in general Hilbert spaces
• Part II: Gabor frames and wavelet frames in L2 (R)
• Part III: Research topics related to frames and operator theory
What the talk really is about: Unification!
(DTU Mathematics)
Talk, Concentration Week, Texas A & M
July 16, 2012
4 / 85
Part I:
Frames in general Hilbert spaces
(DTU Mathematics)
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July 16, 2012
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Plan for the first part of the talk
• Frames and Riesz bases in general Hilbert spaces;
• Dual pairs of frames in general Hilbert spaces H: expansions
f =
(DTU Mathematics)
X
hf , gk ifk , f ∈ H.
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July 16, 2012
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Goal and scope
Let (H, h·, ·i) be a Hilbert space.
Want: Expansions
f =
X
ck fk
of signals f ∈ H in terms of convenient building blocks fk .
(DTU Mathematics)
Talk, Concentration Week, Texas A & M
July 16, 2012
7 / 85
Goal and scope
Let (H, h·, ·i) be a Hilbert space.
Want: Expansions
f =
X
ck fk
of signals f ∈ H in terms of convenient building blocks fk .
Desirable properties could be:
• Easy to calculate the coefficients ck
• Only few large coefficients ck for the relevant signals f (as for wavelet
ONB’s!).
(DTU Mathematics)
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ONB’s
Recall: If {ek }∞
k=1 is an orthonormal basis for H, then each f ∈ H has an
unconditionally convergent expansion
f =
∞
X
k=1
hf , ek iek .
ONB’s are good - but the conditions quite restrictive!
(DTU Mathematics)
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July 16, 2012
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Riesz sequences
Definition: A sequence {fk }∞
k=1 of elements in H is called a Riesz sequence if
there exist constants A, B > 0 such that
2
X
X
X
A
|ck |2 ≤ ck fk ≤ B
|ck |2
for all finite sequences {ck }.
∞
A Riesz sequence {fk }∞
k=1 for which span{fk }k=1 = H is called a Riesz basis.
(DTU Mathematics)
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Riesz sequences
Definition: A sequence {fk }∞
k=1 of elements in H is called a Riesz sequence if
there exist constants A, B > 0 such that
2
X
X
X
A
|ck |2 ≤ ck fk ≤ B
|ck |2
for all finite sequences {ck }.
∞
A Riesz sequence {fk }∞
k=1 for which span{fk }k=1 = H is called a Riesz basis.
Key properties:
• The Riesz bases are precisely the sequences which have the form
∞
∞
{fk }∞
k=1 = {Uek }k=1 , where {ek }k=1 is an orthonormal basis for H and
U : H → H is a bounded bijective operator.
P
1
• ||U −1
||f ||2 ≤ |hf , fk i|2 ≤ ||U||2 ||f ||2 , ∀f ∈ H.
||2
• Letting gk := (U ∗ )−1 fk ,
f =
∞
X
hf , gk ifk , ∀f ∈ H.
k=1
(DTU Mathematics)
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Frames
Definition: A sequence {fk } in H is a frame if there exist constants A, B > 0
such that
X
A ||f ||2 ≤
|hf , fk i|2 ≤ B ||f ||2 , ∀f ∈ H.
A frame is tight if we can take A = B = 1.
The sequence {fk } is a Bessel sequence if at least the upper inequality holds.
(DTU Mathematics)
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Frames
Definition: A sequence {fk } in H is a frame if there exist constants A, B > 0
such that
X
A ||f ||2 ≤
|hf , fk i|2 ≤ B ||f ||2 , ∀f ∈ H.
A frame is tight if we can take A = B = 1.
The sequence {fk } is a Bessel sequence if at least the upper inequality holds.
Given a frame {fk }∞
k=1 , the pre-frame operator or synthesis operator is
T : H → ℓ2 (N), T{ck }∞
k=1 =
∞
X
ck fk .
k=1
Note that
T ∗ : H → ℓ2 (N), T ∗ f = {hf , fk i}∞
k=1 .
(DTU Mathematics)
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Frames versus Riesz bases
• Any Riesz basis is a frame.
(DTU Mathematics)
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Frames versus Riesz bases
• Any Riesz basis is a frame.
∞
• A frame {fk }∞
k=1 is a Riesz basis if and only if {fk }k=1 is a basis.
(DTU Mathematics)
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Frames versus Riesz bases
• Any Riesz basis is a frame.
∞
• A frame {fk }∞
k=1 is a Riesz basis if and only if {fk }k=1 is a basis.
• A frame {fk }∞
k=1 is a Riesz basis if and only if
∞
X
k=1
(DTU Mathematics)
ck fk = 0 ⇒ ck = 0, ∀k.
Talk, Concentration Week, Texas A & M
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Frames versus Riesz bases
• Any Riesz basis is a frame.
∞
• A frame {fk }∞
k=1 is a Riesz basis if and only if {fk }k=1 is a basis.
• A frame {fk }∞
k=1 is a Riesz basis if and only if
∞
X
k=1
ck fk = 0 ⇒ ck = 0, ∀k.
• A frame which is not a Riesz basis, is said to be overcomplete or
redundant: In that case there exists {ck }∞
k=1 \ {0} such that
∞
X
ck fk = 0.
k=1
(DTU Mathematics)
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The frame decomposition
Frame operator associated to frame {fk }∞
k=1 with pre-frame operator T :
S : H → H, Sf = TT ∗ f =
X
hf , fk ifk .
k∈I
Frame decomposition:
f =
X
k∈I
(DTU Mathematics)
hf , S−1 fk ifk , ∀f ∈ H.
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The frame decomposition
Frame operator associated to frame {fk }∞
k=1 with pre-frame operator T :
S : H → H, Sf = TT ∗ f =
X
hf , fk ifk .
k∈I
Frame decomposition:
f =
X
k∈I
hf , S−1 fk ifk , ∀f ∈ H.
Might be difficult to compute!
Frame decomposition for tight frame:
f =
1 X
hf , fk ifk , ∀f ∈ H.
A
k∈I
(DTU Mathematics)
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General dual frames
A frame which is not a Riesz basis is said to be overcomplete.
Theorem: Assume that {fk }∞
k=1 is an overcomplete frame. Then there exist
frames
−1
∞
{gk }∞
k=1 6= {S fk }k=1
for which
f =
∞
X
hf , gk ifk , ∀f ∈ H.
k=1
(DTU Mathematics)
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General dual frames
A frame which is not a Riesz basis is said to be overcomplete.
Theorem: Assume that {fk }∞
k=1 is an overcomplete frame. Then there exist
frames
−1
∞
{gk }∞
k=1 6= {S fk }k=1
for which
f =
∞
X
hf , gk ifk , ∀f ∈ H.
k=1
∞
{gk }∞
k=1 is called a dual frame of {fk }k=1 . The special choice
−1
∞
{gk }∞
k=1 = {S fk }k=1
is called the canonical dual frame.
(DTU Mathematics)
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General dual frames
Note: Let {fk }∞
k=1 be a Bessel sequence with pre-frame operator
T : H → ℓ2 (N), T{ck }∞
k=1 =
∞
X
ck fk
k=1
∞
and {gk }∞
k=1 be a Bessel sequence with pre-frame operator U. Then {fk }k=1
∞
and {gk }k=1 are dual frames if and only if
∞
X
f =
hf , gk ifk , ∀f ∈ H,
k=1
i.e., if and only if
TU ∗ = I,
i.e., if and only if
UT ∗ = I.
(DTU Mathematics)
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Characterization of all dual frames
Result by Shidong Li, 1991:
Theorem: Let {fk }∞
k=1 be a frame with pre-frame operator T. The bounded
2
operators U : ℓ (N) → H for which
UT ∗ = I,
i.e., the bounded left-inverses of T ∗ , are precisely the operators having the
form
U = S−1 T + W(I − T ∗ S−1 T),
where W : ℓ2 (N) → H is a bounded operator and I denotes the identity
operator on ℓ2 (N).
(DTU Mathematics)
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Characterization of all dual frames
Result by Shidong Li, 1991:
∞
Theorem: Let {fk }∞
k=1 be a frame for H. The dual frames of {fk }k=1 are
precisely the families

∞
∞


X
−1
−1
{gk }∞
=
S
f
+
h
−
hS
f
,
f
ih
,
k
k
k
j
j
k=1


j=1
(1)
k=1
where {hk }∞
k=1 is a Bessel sequence in H.
(DTU Mathematics)
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July 16, 2012
16 / 85
Characterization of all dual frames
Result by Shidong Li, 1991:
∞
Theorem: Let {fk }∞
k=1 be a frame for H. The dual frames of {fk }k=1 are
precisely the families

∞
∞


X
−1
−1
{gk }∞
=
S
f
+
h
−
hS
f
,
f
ih
,
k
k
k
j
j
k=1


j=1
(1)
k=1
where {hk }∞
k=1 is a Bessel sequence in H.
Allows us to optimize the duals:
• Which dual has the best approximation theoretic properties?
• Which dual has the smallest support?
• Which dual has the most convenient expression?
• Can we find a dual that is easy to calculate?
(DTU Mathematics)
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An example: Sigma-Delta quantization
Work by Lammers, Powell, and Yilmaz:
Consider a frame {fk }Nk=1 for Rd . Letting {gk }Nk=1 denote a dual frame, each
f ∈ Rd can be written
f =
N
X
k=1
(DTU Mathematics)
hf , gk ifk .
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An example: Sigma-Delta quantization
Work by Lammers, Powell, and Yilmaz:
Consider a frame {fk }Nk=1 for Rd . Letting {gk }Nk=1 denote a dual frame, each
f ∈ Rd can be written
f =
N
X
k=1
hf , gk ifk .
In practice: the coefficients hf , gk i must be quantized, i.e., replaced by some
coefficients dk from a discrete set such that
dk ≈ hf , gk i,
which leads to
f ≈
N
X
dk fk .
k=1
Note: increased redundancy (large N) increases the chance of a good
approximation.
(DTU Mathematics)
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An example: Sigma-Delta quantization
• For each r ∈ N there is a procedure (rth order sigma-delta quantization)
to find appropriate coefficients dk .
(DTU Mathematics)
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An example: Sigma-Delta quantization
• For each r ∈ N there is a procedure (rth order sigma-delta quantization)
to find appropriate coefficients dk .
• rthe order sigma-delta quantization with the canonical dual frame does
not provide approximation order N −r .
• Approximation order N −r can be obtained using other dual frames.
(DTU Mathematics)
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Tight frames versus dual pairs
• For some years: focus on construction of tight frame.
(DTU Mathematics)
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Tight frames versus dual pairs
• For some years: focus on construction of tight frame.
• Do not forget the extra flexibility offered by convenient dual frame pairs!
(DTU Mathematics)
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Tight frames versus dual pairs
• For some years: focus on construction of tight frame.
• Do not forget the extra flexibility offered by convenient dual frame pairs!
Theorem: For each Bessel sequence {fk }∞
k=1 in a Hilbert space H, there exist
such
that
a family of vectors {gk }∞
k=1
∞
{fk }∞
k=1 ∪ {gk }k=1
is a tight frame for H.
(DTU Mathematics)
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Tight frames versus dual pairs
10
Example Let {ej }10
j=1 be an orthonormal basis for C and consider the frame
10
{fj }10
j=1 := {2e1 } ∪ {ej }j=2 .
There exist 9 vectors {hj }9j=1 such that
9
{fj }10
j=1 ∪ {hj }j=1
is a tight frame for C10 - and 9 is the minimal number to add.
A pair of dual frames can be obtained by adding just one element:
10
{fj }10
j=1 ∪ {−3e1 } and {fj }j=1 ∪ {e1 }
form dual frames in C10 .
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem (Casazza and Fickus): Given a sequence of positive numbers
N
a1 ≥ a2 ≥ · · · ≥ aM , there exists a tight frame {fj }M
j=1 for R with
||fj || = aj , j = 1, . . . , M, if and only if
a21 ≤
(DTU Mathematics)
M
1X 2
aj .
N
(2)
j=1
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Tight frames versus dual pairs
Theorem (Casazza and Fickus): Given a sequence of positive numbers
N
a1 ≥ a2 ≥ · · · ≥ aM , there exists a tight frame {fj }M
j=1 for R with
||fj || = aj , j = 1, . . . , M, if and only if
a21 ≤
M
1X 2
aj .
N
(2)
j=1
Theorem (C., Powell, Xiao, 2010): Given any sequence {αj }M
j=1 of real
numbers, and assume that M > N. Then the following are equivalent:
N
e M
(i) There exist a pair of dual frames {fj }M
j=1 and {fj }j=1 for R such that
αj = hfj , efj i for all j = 1, . . . , M.
P
(ii) N = M
j=1 αj .
(DTU Mathematics)
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Part II:
Gabor frames and wavelet frames in L2(R)
(DTU Mathematics)
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Operators on L2 (R)
Translation by a ∈ R: Ta : L2 (R) → L2 (R), (Ta f )(x) = f (x − a).
Modulation by b ∈ R : Eb : L2 (R) → L2 (R), (Eb f )(x) = e2πibx f (x).
Dilation by a > 0 : Da : L2 (R) → L2 (R), (Da f )(x) =
Dyadic scaling:
√1 f ( x ).
a a
D : L2 (R) → L2 (R), (Df )(x) = 21/2 f (2x).
All these operators are unitary on L2 (R).
Important commutator relations:
Ta Eb = e−2πiba Eb Ta , Tb Da = Da Tb/a , Da Eb = Eb/a Da
(DTU Mathematics)
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The Fourier transform
For f ∈ L1 (R), the Fourier transform is defined by
Z ∞
Ff (γ) = f̂ (γ) :=
f (x)e−2πixγ dx, γ ∈ R.
−∞
The Fourier transform can be extended to a unitary operator on L2 (R).
Plancherel’s equation:
hf̂ , ĝi = hf , gi, ∀f , g ∈ L2 (R), and ||f̂ || = ||f ||.
Important commutator relations:
FTa = E−a F, FEa = Ta F,
FDa = D1/a F, FD = D−1 F.
(DTU Mathematics)
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Splines
The B-splines BN , N ∈ N, are given by
B1 = χ[0,1] , BN+1 = BN ∗ B1 .
Theorem: Given N ∈ N, the B-spline BN has the following properties:
(i) supp BN = [0, N] and BN > 0 on ]0, N[.
R∞
(ii) −∞ BN (x)dx = 1.
P
(iii)
k∈Z BN (x − k) = 1
(iv) For any N ∈ N,
cN (γ) =
B
(DTU Mathematics)
1 − e−2πiγ
2πiγ
N
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.
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Splines
1
K2 K1
0
1
1
2
3
4
K2 K1
0
1
2
3
4
Figure: The B-splines B2 and B3 .
(DTU Mathematics)
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Classical wavelet theory
• Given a function ψ ∈ L2 (R) and j, k ∈ Z, let
ψj,k (x) := 2j/2 ψ(2j x − k), x ∈ R.
• In terms of the operators Tk f (x) = f (x − k) and Df (x) = 21/2 f (2x),
ψj,k = Dj Tk ψ, j, k ∈ Z.
• If {ψj,k }j,k∈Z is an orthonormal basis for L2 (R), the function ψ is called a
wavelet.
(DTU Mathematics)
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Multiresolution analysis - a tool to construct a wavelet
Definition: A multiresolution analysis for L2 (R) consists of a sequence of
closed subspaces {Vj }j∈Z of L2 (R) and a function φ ∈ V0 , such that the
following conditions hold:
(i) · · · V−1 ⊂ V0 ⊂ V1 · · · .
(ii) ∪j Vj = L2 (R) and ∩j Vj = {0}.
(iii) f ∈ Vj ⇔ [x → f (2x)] ∈ Vj+1 .
(iv) f ∈ V0 ⇒ Tk f ∈ V0 , ∀k ∈ Z.
(v) {Tk φ}k∈Z is an orthonormal basis for V0 .
(DTU Mathematics)
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Construction of wavelet ONB
• The function φ in a multiresolution analysis satisfies a scaling equation,
φ̂(2γ) = H0 (γ)φ̂(γ), a.e. γ ∈ R,
for some 1-periodic function H0 ∈ L2 (0, 1).
• Let
1
H1 (γ) = H0 (γ + )e−2πiγ .
2
• Then the function ψ defined via
ψ̂(2γ) = H1 (γ)φ̂(γ)
generates a wavelet orthonormal basis {Dj Tk ψ}j,k∈Z .
P
P
• Explicitly: if H1 (γ) = k∈Z dk e2πikγ , then ψ(x) = 2 k∈Z dk φ(2x + k).
(DTU Mathematics)
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Construction of wavelet ONB
Theorem: Let φ ∈ L2 (R), and let
Vj := span{Dj Tk φ}k∈Z .
Assume that the following conditions hold:
(i) infγ∈]−ǫ,ǫ[ |φ̂(γ)| > 0 for some ǫ > 0;
(ii) The scaling equation
φ̂(2γ) = H0 (γ)φ̂(γ),
is satisfied for a bounded 1-periodic function H0 ;
(iii) {Tk φ}k∈Z is an orthonormal system.
Then φ generates a multiresolution analysis, and there exists a wavelet ψ of
the form
X
ψ(x) = 2
dk φ(2x + k).
k∈Z
(DTU Mathematics)
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Spline wavelets
• The B-splines BN , N ∈ N, are given by
B1 = χ[0,1] , BN+1 = BN ∗ B1 .
• One can consider any order splines BN and define associated
multiresolution analyses, which leads to wavelets of the type
X
ψ(x) =
ck BN (2x + k).
k∈Z
• These wavelets are called Battle–Lemarié wavelets.
• Only shortcoming: except for the case N = 1, all coefficients ck are
non-zero, which implies that the wavelet ψ has support equal to R.
(DTU Mathematics)
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Spline wavelets - can we do better for N > 1?
(DTU Mathematics)
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Spline wavelets - can we do better for N > 1?
Can show:
• There does not exists an ONB {Dj Tk ψ}j,k∈Z for L2 (R) generated by a
finite linear combination
ψ(x) =
(DTU Mathematics)
X
ck BN (2x + k).
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Spline wavelets - can we do better for N > 1?
Can show:
• There does not exists an ONB {Dj Tk ψ}j,k∈Z for L2 (R) generated by a
finite linear combination
ψ(x) =
X
ck BN (2x + k).
• There does not exists a tight frame {Dj Tk ψ}j,k∈Z for L2 (R) generated by
a finite linear combination
ψ(x) =
(DTU Mathematics)
X
ck BN (2x + k).
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Spline wavelets - can we do better for N > 1?
Can show:
• There does not exists an ONB {Dj Tk ψ}j,k∈Z for L2 (R) generated by a
finite linear combination
ψ(x) =
X
ck BN (2x + k).
• There does not exists a tight frame {Dj Tk ψ}j,k∈Z for L2 (R) generated by
a finite linear combination
ψ(x) =
X
ck BN (2x + k).
• There does not exists a pairs of dual wavelet frames {Dj Tk ψ}j,k∈Z and
{Dj Tk ψ̃}j,k∈Z for which ψ and ψ̃ are finite linear combinations of
functions DTk BN , j, k ∈ Z.
(DTU Mathematics)
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Spline wavelets
Solution: consider systems of the wavelet-type, but generated by more than
one function.
Setup for construction of tight wavelet frames by Ron and Shen:
Let ψ0 ∈ L2 (R) and assume that
(i) There exists a function H0 ∈ L∞ (T) such that
(ii) limγ→0 ψb0 (γ) = 1.
ψb0 (2γ) = H0 (γ)ψb0 (γ).
Further, let H1 , . . . , Hn ∈ L∞ (T), and define ψ1 , . . . , ψn ∈ L2 (R) by
cℓ (2γ) = Hℓ (γ)ψb0 (γ), ℓ = 1, . . . , n.
ψ
(DTU Mathematics)
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The unitary extension principle
• We want to find conditions on the functions H1 , . . . , Hn such that
ψ1 , . . . , ψn generate a multiwavelet frame for L2 (R).
• Let H denote the (n + 1) × 2 matrix-valued function defined by



H(γ) = 


(DTU Mathematics)
H0 (γ) T1/2 H0 (γ)
H1 (γ) T1/2 H1 (γ)
·
·
·
·
Hn (γ) T1/2 Hn (γ)



 , γ ∈ R.


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The unitary extension principle
Theorem (Ron and Shen, 1997): Let {ψℓ , Hℓ }nℓ=0 be as in the general setup,
and assume that H(γ)∗ H(γ) = I for a.e. γ ∈ T. Then the multiwavelet system
{Dj Tk ψℓ }j,k∈Z,ℓ=1,...,n constitutes a tight frame for L2 (R) with frame bound
equal to 1.
Can be applied to any order B-spline!
1
K2
K1
0
1
1
2
K2
K1
K1
0
1
2
K1
Figure: The two wavelet frame generators ψ1 and ψ2 associated with ψ0 = B2 .
(DTU Mathematics)
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Gabor systems
Gabor systems: have the form
{e2πimbx g(x − na)}m,n∈Z
for some g ∈ L2 (R), a, b > 0. Short notation:
{Emb Tna g}m,n∈Z = {e2πimbx g(x − na)}
(DTU Mathematics)
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Gabor systems
Gabor systems: have the form
{e2πimbx g(x − na)}m,n∈Z
for some g ∈ L2 (R), a, b > 0. Short notation:
{Emb Tna g}m,n∈Z = {e2πimbx g(x − na)}
Example:
• {e2πimx χ[0,1] (x)}m∈Z is an ONB for L2 (0, 1)
• For n ∈ Z, {e2πim(x−n) χ[0,1] (x − n)}m∈Z = {e2πimx χ[0,1] (x − n)}m∈Z is
an ONB for L2 (n, n + 1)
• {e2πimx χ[0,1] (x − n)}m,n∈Z is an ONB for L2 (R)
(DTU Mathematics)
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Gabor frames and Riesz bases
(DTU Mathematics)
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Gabor frames and Riesz bases
• If {Emb Tna g}m,n∈Z is a frame, then ab ≤ 1;
(DTU Mathematics)
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Gabor frames and Riesz bases
• If {Emb Tna g}m,n∈Z is a frame, then ab ≤ 1;
• If {Emb Tna g}m,n∈Z is a frame, then
{fk }∞
k=1 is a Riesz basis ⇔ ab = 1.
(DTU Mathematics)
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Gabor frames and Riesz bases
• If {Emb Tna g}m,n∈Z is a frame, then ab ≤ 1;
• If {Emb Tna g}m,n∈Z is a frame, then
{fk }∞
k=1 is a Riesz basis ⇔ ab = 1.
For the sake of time-frequency analysis: we want the Gabor frame
{Emb Tna g}m,n∈Z to be generated by a continuous function g with compact
support.
(DTU Mathematics)
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Gabor systems
Lemma: If g is be a continuous function with compact support, then
• {Emb Tna g}m,n∈Z can not be an ONB.
(DTU Mathematics)
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Gabor systems
Lemma: If g is be a continuous function with compact support, then
• {Emb Tna g}m,n∈Z can not be an ONB.
• {Emb Tna g}m,n∈Z can not be a Riesz basis.
(DTU Mathematics)
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Gabor systems
Lemma: If g is be a continuous function with compact support, then
• {Emb Tna g}m,n∈Z can not be an ONB.
• {Emb Tna g}m,n∈Z can not be a Riesz basis.
• {Emb Tna g}m,n∈Z can be a frame if 0 < ab < 1;
Thus, it is necessary to consider frames if we want Gabor systems with good
properties.
(DTU Mathematics)
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Pairs of dual Gabor frames
Two Bessel sequences {Emb Tna g}m,n∈Z and {Emb Tna h}m,n∈Z form dual frames
if
X
f =
hf , Emb Tna hiEmb Tna g, ∀f ∈ L2 (R).
m,n∈Z
Ron & Shen, A.J.E.M. Janssen (1998):
Theorem: Two Bessel sequences {Emb Tna g}m,n∈Z and {Emb Tna h}m,n∈Z form
dual frames if and only if
X
g(x − n/b − ka)h(x − ka) = bδn,0 , a.e. x ∈ [0, a].
k∈Z
(DTU Mathematics)
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History
• Bölsckei, Janssen, 1998-2000: For b rational, characterization of Gabor
frames with compactly supported window having a compactly supported
dual window;
• Feichtinger, Gröchenig (1997): window in S0 implies that the canonical
dual window is in S0 ;
• Krishtal, Okoudjou, 2007: window in W(L∞ , ℓ1 ) implies that the
canonical dual window is in W(L∞ , ℓ1 ).
(DTU Mathematics)
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Duality principle
The duality principle (Janssen, Daubechies, Landau, and Landau, and Ron
and Shen: concerns the relationship between frame properties for a function g
with respect to the lattice {(na, mb)}m,n∈Z and with respect to the so-called
dual lattice {(n/b, m/a)}m,n∈Z :
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the following are
equivalent:
(i) {Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B;
(ii) { √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
(DTU Mathematics)
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Duality principle
The duality principle (Janssen, Daubechies, Landau, and Landau, and Ron
and Shen: concerns the relationship between frame properties for a function g
with respect to the lattice {(na, mb)}m,n∈Z and with respect to the so-called
dual lattice {(n/b, m/a)}m,n∈Z :
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the following are
equivalent:
(i) {Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B;
(ii) { √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
Intuition: If {Emb Tna g}m,n∈Z is a frame for L2 (R), then ab ≤ 1, i.e., the
sampling points {(na, mb)}m,n∈Z are “sufficiently dense.” Therefore the
points {(n/b, m/a)}m,n∈Z are “sparse,” and therefore { √1ab Em/a Tn/b g}m,n∈Z
are linearly independent and only span a subspace of L2 (R).
(DTU Mathematics)
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Wexler-Raz’ Theorem
Wexler-Raz’ Theorem: If the Gabor systems {Emb Tna g}m,n∈Z and
{Emb Tna h}m,n∈Z are dual frames, then the Gabor systems
{ √1ab Em/a Tn/b g}m,n∈Z and { √1ab Em/a Tn/b h}m,n∈Z are biorthogonal, i.e.,
1
1
h √ Em/a Tn/b g, √ Em′ /a Tn′ /b hi = δm,m′ δn,n′ .
ab
ab
(DTU Mathematics)
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Explicit construction of dual pairs of Gabor frames
∞
In order for a frame {fk }∞
k=1 to be useful, we need a dual frame {gk }k=1 , i.e.,
a frame such that
f =
∞
X
hf , gk ifk , ∀f ∈ H.
k=1
How can we construct convenient dual frames?
Ansatz/suggestion: Given a window function g ∈ L2 (R)generating a frame
{Emb Tna g}m,n∈Z , look for a dual window of the form
h(x) =
K
X
ck g(x + k).
k=−K
The structure of h makes it easy to derive properties of h based on properties
of g (regularity, size of support, membership un various vector spaces,....)
(DTU Mathematics)
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Explicit construction of dual pairs of Gabor frames
Theorem:(C., 2006) Let N ∈ N. Let g ∈ L2 (R) be a real-valued bounded
function for which
• supp g ⊆ [0, N],
•
P
Let b
n∈Z g(x − n) =
1
]. Then
∈]0, 2N−1
1.
the function g and the function h defined by
h(x) = bg(x) + 2b
N−1
X
g(x + n)
n=1
generate dual frames {Emb Tn g}m,n∈Z and {Emb Tn h}m,n∈Z for L2 (R).
(DTU Mathematics)
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Candidates for g - the B-splines
For the B-spline case:
• The functions BN and the dual window
h(x) = bBN (x) + 2b
N−1
X
BN (x + n)
n=1
are splines;
• BN and h have compact support, i.e., perfect time–localization;
cN and h of any
• By choosing N sufficiently large, polynomial decay of B
desired order can be obtained.
Note:
(DTU Mathematics)
cN (γ) =
B
sin πγ
πγ
N
e−πiNγ .
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Example
For the B-spline


x
B2 (x) = 2 − x


0
x ∈ [0, 1[,
x ∈ [1, 2[,
x∈
/ [0, 2[,
we can use the result for b ∈]0, 1/3]. For b = 1/3 we obtain the dual
generator
1
2
B2 (x) + B2 (x + 1)
3
3

2

x ∈ [−1, 0[,
 3 (x + 1)
1
=
x ∈ [0, 2[,
3 (2 − x)


0
x∈
/ [−1, 2[.
h(x) =
(DTU Mathematics)
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Example
1
K3 K2 K1
0
1
1
2
3
4
K3 K2 K1
0
1
2
3
4
Figure: The B-spline N2 and the dual generator h for b = 1/3; and the B-spline N3
and the dual generator h with b = 1/5.
(DTU Mathematics)
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Other choices? Yes!
Theorem: (C., Kim, 2007) Let N ∈ N. Let g ∈ L2 (R) be a real-valued
bounded function for which
• supp g ⊆ [0, N],
•
P
Let b
n∈Z g(x − n) = 1.
1
]. Define h
∈]0, 2N−1
∈ L2 (R) by
h(x) =
N−1
X
an g(x + n),
n=−N+1
where
a0 = b, an + a−n = 2b, n = 1, 2, · · · , N − 1.
Then g and h generate dual frames {Emb Tn g}m,n∈Z and {Emb Tn h}m,n∈Z for
L2 (R).
(DTU Mathematics)
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Example: B-splines revisited
1) Take
a0 = b, an = 0 for n = −N + 1, . . . , −1, an = 2b, n = 1, . . . N − 1.
This is the previous Theorem. This choice gives the shortest support.
2) Take
a−N+1 = a−N+2 = · · · = aN−1 = b :
if g is symmetric, this leads to a symmetric dual generator
h(x) = b
N−1
X
g(x + n).
n=−N+1
Note: h(x) = b on supp g.
(DTU Mathematics)
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B-splines revisited
1
0.7
0.8
0.6
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.1
0
0
-2
-1
0
1
2
3
4
-2
x
0
2
4
6
x
Figure: The generators B2 and B3 and their dual generators via 2).
(DTU Mathematics)
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Explicit constructions
Another class of examples:
Exponential B-splines: Given a sequence of scalars β1 , β2 , . . . , βN ∈ R, let
EN := eβ1 (·) χ[0,1] (·) ∗ eβ2 (·) χ[0,1] (·) ∗ · · · ∗ eβN (·) χ[0,1] (·).
• The function EN is supported on [0, N].
• If βk = 0 for at least one k = 1, . . . , N, then EN satisfies the partition of
unity condition (up to a constant).
• If βj 6= βk for j 6= k, an explicit expression for EN is known (C., Peter
Massopust, 2010).
(DTU Mathematics)
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Explicit constructions
Assume that βk = (k − 1)β, k = 1, . . . , N. Then
EN (x) =

N−1
X

1

1

eβkx ,
x ∈ [0, 1];

β N−1

N

Y

k=0


(k + 1 − j)




j=1


j6=k+1







X
βj1
βjℓ−1

[e + · · · + e
]




 0≤j1 <···<jℓ−1 ≤N−1



N−1



X  j1 ,...,jℓ−1 6=k−1

 βk(x−ℓ+1) x ∈ [ℓ − 1, ℓ]
(−1)ℓ−1


,
.

e

β N−1
N



Y
ℓ = 2, . . . , N


k=0




(k + 1 − j)






j=1
j6=k+1
(DTU Mathematics)
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Explicit constructions
X
k∈Z
EN (x − k) =
eβm − 1
N−1
Y
m=1
β N−1 (N
− 1)!
.
Via the Theorem: construction of dual Gabor frames with generators
EN ,
(DTU Mathematics)
hN =
N−1
X
k=−N+1
ck EN (x + K).
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From Gabor frames to wavelet frames - duality conditions:
Theorem: Two Bessel sequences {Emb Tna g}m,n∈Z and {Emb Tna h}m,n∈Z form
dual frames if and only if
P
(i)
k∈Z g(x − ka)h(x − ka) = b, a.e. x ∈ [0, a].
P
(ii)
k∈Z g(x − n/b − ka)h(x − ka) = 0, a.e. x ∈ [0, a], n ∈ Z \ {0}.
(DTU Mathematics)
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From Gabor frames to wavelet frames - duality conditions:
Theorem: Two Bessel sequences {Emb Tna g}m,n∈Z and {Emb Tna h}m,n∈Z form
dual frames if and only if
P
(i)
k∈Z g(x − ka)h(x − ka) = b, a.e. x ∈ [0, a].
P
(ii)
k∈Z g(x − n/b − ka)h(x − ka) = 0, a.e. x ∈ [0, a], n ∈ Z \ {0}.
Theorem: Given a > 1, b > 0, two Bessel sequences {Daj Tkb ψ}j,k∈Z and
e j,k∈Z , where ψ, ψe ∈ L2 (R), form dual wavelet frames for L2 (R) if
{Daj Tkb ψ}
and only if the following two conditions are satisfied:
P
b j be j
(i)
j∈Z ψ(a γ)ψ(a γ) = b for a.e. γ ∈ R.
(ii) For any number α 6= 0 of the form α = m/aj , m, j ∈ Z,
X
{(j,m)∈Z2 | α=m/aj }
(DTU Mathematics)
be j
b j γ)ψ(a
ψ(a
γ + m/b) = 0, a.e. γ ∈ R.
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From Gabor frames to wavelet frames
C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet
frames based on dual pairs of Gabor frames.
(DTU Mathematics)
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From Gabor frames to wavelet frames
C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet
frames based on dual pairs of Gabor frames.
Let θ > 1 be given. Associated with a function g ∈ L2 (R) with the property
that g(logθ | · |) ∈ L2 (R) we define a function ψ ∈ L2 (R) by
b
ψ(γ)
= g(logθ (|γ|)).
Then
b j γ) = g(j log (a) + log (|γ|)).
ψ(a
θ
θ
(DTU Mathematics)
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From Gabor frames to wavelet frames
C., Say Song Goh, 2012: metod for construction of dual pairs of wavelet
frames based on dual pairs of Gabor frames.
Let θ > 1 be given. Associated with a function g ∈ L2 (R) with the property
that g(logθ | · |) ∈ L2 (R) we define a function ψ ∈ L2 (R) by
b
ψ(γ)
= g(logθ (|γ|)).
Then
b j γ) = g(j log (a) + log (|γ|)).
ψ(a
θ
θ
When applied to exponential B-splines:
e for
Construction of dual pairs of wavelet frames with generators ψ and ψ,
which
be
b and ψ
• ψ
are compactly supported splines with geometrically distributed
knot sequences.
(DTU Mathematics)
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Example
The exponential B-spline E2 with β1 = 0, β2 = 1 :


0,
x∈
/ [0, 2],






E2 (x) = ex − 1,
x ∈ [0, 1],







e − e−1 ex , x ∈ [1, 2].
Then
X
k∈Z
E2 (x − k) = e − 1, x ∈ R,
so we consider the function
g(x) := (e − 1)−1 E2 (x).
(DTU Mathematics)
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Example
• Let ψ be defined via
b
ψ(γ)
=

0,







|γ| ∈
/ [1, e2 ],
|γ|−1
e−1 ,






 e−e−1 |γ|
e−1
|γ| ∈ [1, e],
, |γ| ∈ [e, e2 ].
b is a geometric spline with knots at the points ±1, ±e, ±e2 .
• ψ
e defined by
• Let b = 15−1 . Then the function ψ
1
1 X b n
be
ψ(|e γ|)
ψ(γ)
=
15
n=−1
is a dual generator.
be
• ψ
is a geometric spline with knots at ±e−1 , ±1, ±e2 , ±e3 .
(DTU Mathematics)
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Example
be
Figure: Plots of the geometric splines ψb and ψ.
(DTU Mathematics)
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From to wavelet frames to Gabor frames
The Meyer wavelet is the function ψ ∈ L2 (R) defined via

π
iπγ

if 1/3 ≤ |γ| ≤ 2/3,
e sin( 2 (ν(3|γ| − 1))),
π
iπγ
b
ψ(γ) = e cos( 2 (ν(3|γ|/2 − 1))), if 2/3 ≤ |γ| ≤ 4/3,


0,
if |γ| ∈
/ [1/3, 4/3],
where ν : R → R is any continuous function for which
(
0, if x ≤ 0,
ν(x) =
1, if x ≥ 1,
and
ν(x) + ν(1 − x) = 1, x ∈ R.
Known: {D2j Tk ψ}j,k∈Z is an orthonormal basis for L2 (R), and
(DTU Mathematics)
supp ψb = [−4/3, −1/3] ∪ [1/3, 4/3].
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From to wavelet frames to Gabor frames
Let

h
i
−1

, if 0 < x < 1,
 exp − {exp [x/(1 − x)] − 1}
ν0 (x) =
0,
if x ≤ 0,


1,
if x ≥ 1,
ν(x) :=
1
(ν0 (x) − ν0 (1 − x) + 1), x ∈ R.
2

π
x

√1

 2 sin( 2 (ν(3 · 2 − 1))),
τ (x) := √1 cos( π2 (ν( 32 · 2x − 1))),
2


0,
3
if − ln
ln 2 ≤ x ≤ 1 −
if 1
if x
ln 3
ln 2 ,
3
ln 3
− ln
ln 2 ≤ x ≤ 2 − ln 2 ,
3
ln 3
∈
/ [− ln
ln 2 , 2 − ln 2 ],
Then τ is real-valued, compactly supported, belongs to C∞ (R), and
{Em/2 Tn τ }m,n∈Z is a tight frame with bound A = 1.
(DTU Mathematics)
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From to wavelet frames to Gabor frames
Figure: The function τ, which is C∞ and has compact support. The Gabor system
{Em/2 Tn τ }m,n∈Z is a tight frame with bound A = 1.
(DTU Mathematics)
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Part III:
Research problems related to frames and
operator theory
(DTU Mathematics)
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Open problems
• An extension problem for wavelet frames
• The duality principle in general Hilbert spaces
(DTU Mathematics)
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An extension problem for wavelet frames
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem: For each Bessel sequence {fk }∞
k=1 in a Hilbert space H, there exist
such
that
a family of vectors {gk }∞
k=1
∞
{fk }∞
k=1 ∪ {gk }k=1
is a tight frame for H.
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem: For each Bessel sequence {fk }∞
k=1 in a Hilbert space H, there exist
such
that
a family of vectors {gk }∞
k=1
∞
{fk }∞
k=1 ∪ {gk }k=1
is a tight frame for H.
Theorem (D. Li and W.Sun, 2009): Let {Emb Tna g1 }m,n∈Z be Bessel sequences
in L2 (R), and assume that ab ≤ 1. Then the following hold:
• There exists a Gabor systems {Emb Tna g2 }m,n∈Z such that
{Emb Tna g1 }m,n∈Z ∪ {Emb Tna g2 }m,n∈Z
is a tight frame for L2 (R).
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem: For each Bessel sequence {fk }∞
k=1 in a Hilbert space H, there exist
such
that
a family of vectors {gk }∞
k=1
∞
{fk }∞
k=1 ∪ {gk }k=1
is a tight frame for H.
Theorem (D. Li and W.Sun, 2009): Let {Emb Tna g1 }m,n∈Z be Bessel sequences
in L2 (R), and assume that ab ≤ 1. Then the following hold:
• There exists a Gabor systems {Emb Tna g2 }m,n∈Z such that
{Emb Tna g1 }m,n∈Z ∪ {Emb Tna g2 }m,n∈Z
is a tight frame for L2 (R).
• If g1 has compact support and |suppg1 | ≤ b−1 , then g2 can be chosen to
have compact support.
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem (C., Kim, Kim, 2011): Let {Emb Tna g1 }m,n∈Z and {Emb Tna h1 }m,n∈Z
be Bessel sequences in L2 (R), and assume that ab ≤ 1. Then the following
hold:
• There exist Gabor systems {Emb Tna g2 }m,n∈Z and {Emb Tna h2 }m,n∈Z in
L2 (R) such that
{Emb Tna g1 }m,n∈Z ∪ {Emb Tna g2 }m,n∈Z and {Emb Tna h1 }m,n∈Z ∪ {Emb Tna h2 }m
form a pair of dual frames for L2 (R).
• If g1 and h1 have compact support, the functions g2 and h2 can be chosen
to have compact support.
(DTU Mathematics)
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Tight frames versus dual pairs
Theorem (C., Kim, Kim, 2011): Let {Emb Tna g1 }m,n∈Z and {Emb Tna h1 }m,n∈Z
be Bessel sequences in L2 (R), and assume that ab ≤ 1. Then the following
hold:
• There exist Gabor systems {Emb Tna g2 }m,n∈Z and {Emb Tna h2 }m,n∈Z in
L2 (R) such that
{Emb Tna g1 }m,n∈Z ∪ {Emb Tna g2 }m,n∈Z and {Emb Tna h1 }m,n∈Z ∪ {Emb Tna h2 }m
form a pair of dual frames for L2 (R).
• If g1 and h1 have compact support, the functions g2 and h2 can be chosen
to have compact support.
Note: closely related to work by Han (2009), where it is assumed that
{Emb Tna g1 }m,n∈Z and {Emb Tna h1 }m,n∈Z are dual frames for a subspace.
(DTU Mathematics)
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The wavelet case
f1 }j,k∈Z be
Theorem (C., Kim, Kim, 2011): Let {Dj Tk ψ1 }j,k∈Z and {Dj Tk ψ
2
f1 satisfies
Bessel sequences in L (R). Assume that the Fourier transform of ψ
c
f1 ⊆ [−1, 1].
supp ψ
f2 }j,k∈Z such that
Then there exist wavelet systems {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
f1 }j,k∈Z ∪ {Dj Tk ψ
f2 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
form dual frames for L2 (R).
c1 is
Corollary: (C., Kim, Kim, 2011): In the above setup, assume that ψ
compactly supported and that
c
f1 ⊆ [−1, 1] \ [−ǫ, ǫ]
supp ψ
f2 can be chosen to have
for some ǫ > 0. Then the functions ψ2 and ψ
compactly supported Fourier transforms as well.
(DTU Mathematics)
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The wavelet case - Open problems:
(DTU Mathematics)
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The wavelet case - Open problems:
f1 }j,k∈Z be Bessel sequences in L2 (R).
• Let {Dj Tk ψ1 }j,k∈Z and {Dj Tk ψ
c
f1 is NOT contained in [−1, 1]. Does there exist
Assume that supp ψ
f2 }j,k∈Z such that
wavelet systems {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
f1 }j,k∈Z ∪ {Dj Tk ψ
f2 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
form dual frames for L2 (R)?
(DTU Mathematics)
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The wavelet case - Open problems:
f1 }j,k∈Z be Bessel sequences in L2 (R).
• Let {Dj Tk ψ1 }j,k∈Z and {Dj Tk ψ
c
f1 is NOT contained in [−1, 1]. Does there exist
Assume that supp ψ
f2 }j,k∈Z such that
wavelet systems {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
f1 }j,k∈Z ∪ {Dj Tk ψ
f2 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
form dual frames for L2 (R)?
c1 an ψ
c2 are compactly
• If yes (maybe with additional constraints): If ψ
c
f1 is Not a subset of [−1, 1] \ [−ǫ, ǫ] for some ǫ > 0,
supported, but supp ψ
f2 to have compactly supported Fourier
can we choose ψ2 and ψ
transforms?
(DTU Mathematics)
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The wavelet case - Open problems:
f1 }j,k∈Z be Bessel sequences in L2 (R).
• Let {Dj Tk ψ1 }j,k∈Z and {Dj Tk ψ
c
f1 is NOT contained in [−1, 1]. Does there exist
Assume that supp ψ
f2 }j,k∈Z such that
wavelet systems {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
f1 }j,k∈Z ∪ {Dj Tk ψ
f2 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
form dual frames for L2 (R)?
c1 an ψ
c2 are compactly
• If yes (maybe with additional constraints): If ψ
c
f1 is Not a subset of [−1, 1] \ [−ǫ, ǫ] for some ǫ > 0,
supported, but supp ψ
f2 to have compactly supported Fourier
can we choose ψ2 and ψ
transforms?
• Any positive or negative conclusion is welcome!
(DTU Mathematics)
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The wavelet case - Open problems:
f1 }j,k∈Z be Bessel sequences in L2 (R).
• Let {Dj Tk ψ1 }j,k∈Z and {Dj Tk ψ
c
f1 is NOT contained in [−1, 1]. Does there exist
Assume that supp ψ
f2 }j,k∈Z such that
wavelet systems {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
f1 }j,k∈Z ∪ {Dj Tk ψ
f2 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ2 }j,k∈Z and {Dj Tk ψ
form dual frames for L2 (R)?
c1 an ψ
c2 are compactly
• If yes (maybe with additional constraints): If ψ
c
f1 is Not a subset of [−1, 1] \ [−ǫ, ǫ] for some ǫ > 0,
supported, but supp ψ
f2 to have compactly supported Fourier
can we choose ψ2 and ψ
transforms?
• Any positive or negative conclusion is welcome!
Note: A pair of wavelet Bessel sequences can always be extended to dual
wavelet frame pairs by adding two pairs of wavelet systems.
(DTU Mathematics)
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The wavelet case
• Conjecture, Han, 2009: Let {Dj Tk ψ1 }j,k∈Z be a wavelet frame with upper
frame bound B. Then there exists D > B such that for each K ≥ D, there
f1 ∈ L2 (R) such that
exists ψ
f1 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ
is a tight frame for L2 (R) with bound K.
(DTU Mathematics)
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The wavelet case
• Conjecture, Han, 2009: Let {Dj Tk ψ1 }j,k∈Z be a wavelet frame with upper
frame bound B. Then there exists D > B such that for each K ≥ D, there
f1 ∈ L2 (R) such that
exists ψ
f1 }j,k∈Z
{Dj Tk ψ1 }j,k∈Z ∪ {Dj Tk ψ
is a tight frame for L2 (R) with bound K.
• Based on an example, where
c1 ⊆ [−1, 1].
supp ψ
(DTU Mathematics)
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The duality principle in general Hilbert spaces
(DTU Mathematics)
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Gabor systems
Gabor systems: have the form
{e2πimbx g(x − na)}m,n∈Z
for some g ∈ L2 (R), a, b > 0. Short notation:
{Emb Tna g}m,n∈Z = {e2πimbx g(x − na)}
(DTU Mathematics)
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Duality principle, Wexler-Raz’ Theorem
The duality principle:
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the following are
equivalent:
(i) {Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B;
(ii) { √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
Wexler-Raz’ Theorem: If the Gabor systems {Emb Tna g}m,n∈Z and
{Emb Tna h}m,n∈Z are dual frames, then the Gabor systems
{ √1ab Em/a Tn/b g}m,n∈Z and { √1ab Em/a Tn/b h}m,n∈Z are biorthogonal, i.e.,
1
1
h √ Em/a Tn/b g, √ Em′ /a Tn′ /b hi = δm,m′ δn,n′ .
ab
ab
(DTU Mathematics)
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Duality principle
Can the duality principle in Gabor analysis be recast as a special case of a
general theory, valid for general frames?
(DTU Mathematics)
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Abstract duality in a Hilbert space H
R-dual of a sequence {fi }i∈I in a Hilbert space H, introduced by Casazza,
Kutyniok, and Lammers:
Definition: Let {ei }i∈I and {hi }i∈I denote
Porthonormal bases for H, and let
{fi }i∈I be any sequence in H for which i∈I |hfi , ej i|2 < ∞ for all j ∈ I. The
R-dual of {fi }i∈I with respect to the orthonormal bases {ei }i∈I and {hi }i∈I is
the sequence {ωj }j∈I given by
X
hfi , ej ihi , i ∈ I.
(3)
ωj =
i∈I
(DTU Mathematics)
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Abstract duality - results by Casazza, Kutyniok, Lammers:
Theorem: Define the R-dual {ωj }j∈I of a sequence {fi }i∈I as above. Then:
(i) For all i ∈ I,
X
hωj , hi iej ,
(4)
fi =
j∈I
(ii)
(iii)
(iv)
(v)
i.e., {fi }i∈I is the R-dual sequence of {ωj }j∈I w.r.t. the orthonormal bases
{hi }i∈I and {ei }i∈I .
{fi }i∈I is a Bessel sequence if and only {ωi }i∈I is a Bessel sequence.
{fi }i∈I satisfies the lower frame condition with bound A if and only if
{ωj }j∈I satisfies the lower Riesz sequence condition with bound A.
{fi }i∈I is a frame for H with bounds A, B if and only if {ωj }j∈I is a Riesz
sequence in H with bounds A, B.
Two Bessel sequences {fi }i∈I and {gi }i∈I in H are dual frames if and
only if the associated R-dual sequences {ωj }j∈I and {γj }j∈I satisfy that
hωj , γk i = δj,k , j, k ∈ I.
(DTU Mathematics)
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(5)
July 16, 2012
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The duality principle in Gabor analysis
The duality principle:
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the Gabor system
{Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B if and only if
{ √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
(DTU Mathematics)
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The duality principle in Gabor analysis
The duality principle:
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the Gabor system
{Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B if and only if
{ √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
• Can this result be derived as a consequence of the abstract duality
concept?
(DTU Mathematics)
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The duality principle in Gabor analysis
The duality principle:
Theorem: Let g ∈ L2 (R) and a, b > 0 be given. Then the Gabor system
{Emb Tna g}m,n∈Z is a frame for L2 (R) with bounds A, B if and only if
{ √1ab Em/a Tn/b g}m,n∈Z is a Riesz sequence with bounds A, B.
• Can this result be derived as a consequence of the abstract duality
concept?
• That is, can { √1 Em/a Tn/b g}m,n∈Z be realized as the R-dual of
ab
{Emb Tna g}m,n∈Z w.r.t. certain choices of orthonormal bases {ei }i∈I and
{hi }i∈I ?
(DTU Mathematics)
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Abstract duality
Result by Casazza, Kutyniok, Lammers:
• If {Emb Tna g}m,n∈Z is a frame and ab = 1, then { √1 Em/a Tn/b g}m,n∈Z
ab
can be realized as the R-dual of {Emb Tna g}m,n∈Z w.r.t. certain choices of
orthonormal bases {ei }i∈I and {hi }i∈I .
(DTU Mathematics)
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Abstract duality
Result by Casazza, Kutyniok, Lammers:
• If {Emb Tna g}m,n∈Z is a frame and ab = 1, then { √1 Em/a Tn/b g}m,n∈Z
ab
can be realized as the R-dual of {Emb Tna g}m,n∈Z w.r.t. certain choices of
orthonormal bases {ei }i∈I and {hi }i∈I .
• If {Emb Tna g}m,n∈Z is a tight frame, then { √1 Em/a Tn/b g}m,n∈Z can be
ab
realized as the R-dual of {Emb Tna g}m,n∈Z w.r.t. certain choices of
orthonormal bases {ei }i∈I and {hi }i∈I .
(DTU Mathematics)
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Abstract duality - a general approach
Question A: What are the conditions on two sequences {fi }i∈I , {ωj }j∈I such
that {ωj }j∈I is the R-dual of {fi }i∈I with respect to some choice of the
orthonormal bases {ei }i∈I and {hi }i∈I ?
(DTU Mathematics)
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Abstract duality - a general approach
Question A: What are the conditions on two sequences {fi }i∈I , {ωj }j∈I such
that {ωj }j∈I is the R-dual of {fi }i∈I with respect to some choice of the
orthonormal bases {ei }i∈I and {hi }i∈I ?
• We will always assume that {fi }i∈I is a frame for H.
• We arrive at the following equivalent formulation of Question A:
Question B: Let {fi }i∈I be a frame for H and {ωj }j∈I a Riesz sequence in H.
Under what conditions can we find orthonormal bases {ei }i∈I and {hi }i∈I for
H such that
X
hωj , hi iej ,
fi =
j∈I
holds?
(DTU Mathematics)
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Abstract duality
Theorem: (C., Kim, Kim) Let {ωj }j∈I be a Riesz basis for the subspace W of
H, with dual Riesz basis {f
ωk }k∈I . Let {ei }i∈I be an orthonormal basis for H.
Given any sequence {fi }i∈I in H, the following hold:
(i) There exists a sequence {hi }i∈I in H such that
X
fi =
hωj , hi iej , ∀i ∈ I.
(6)
j∈I
(ii) The sequences {hi }i∈I satisfying (6) are characterized as
hi = mi + ni ,
where mi ∈ W ⊥ and
ni :=
(7)
X
hek , fi if
ωk , i ∈ I.
k∈I
Question C: When is it possible to find an orthonormal basis {hi }i∈I for H of
the form (7)?
(DTU Mathematics)
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The duality principle in Gabor analysis
Key question: Given a frame {fi }i∈I and a Riesz sequence {ωj }j∈I , both with
bounds A, B. Let {f
ωk }k∈I denote the dual Riesz basis.
• Can we find an ONB {ei }i∈I such that {ni }i∈I , given by
ni :=
X
hek , fi if
ωk , i ∈ I,
k∈I
is a tight frame with bound 1?
• Or can we find a case where NO choice of {ei }i∈I makes {ni }i∈I a tight
frame with bound 1?
(DTU Mathematics)
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Abstract duality - a partial answer
Corollary: (C., Kim, Kim) In the above setup - if {ωj }j∈I is a Riesz basis for
H, then
fi =
X
hωj , hi iej , ∀i ∈ I
j∈I
has the unique solution
hi = ni , i ∈ I.
Thus, Question C has an affirmative answer if and only if {ni }i∈I is an ONB.
(DTU Mathematics)
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Abstract duality
Theorem: (C., Kim, Kim) Let {ωj }j∈I be a Riesz sequence spanning a proper
subspace W of H and {ei }i∈I an orthonormal basis for H. Given any frame
{fi }i∈I for H, the following are equivalent:
(i) {ωj }j∈I is an R-dual of {fi }i∈I w.r.t. {ei }i∈I and some orthonormal basis
{hi }i∈I .
(ii) There exists an orthonormal basis {hi }i∈I for H satisfying
fi =
X
hωj , hi iej , ∀i ∈ I.
j∈I
(iii) The sequence {ni }i∈I is a tight frame for W with frame bound E = 1.
(DTU Mathematics)
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Abstract duality
Corollary: (C., Kim, Kim) Assume that {ωj }j∈I is a tight orthogonal Riesz
sequence spanning a proper subspace of H with (Riesz) bound A and that
{fi }i∈I is a tight frame for H with frame bound A. Then the following hold:
(i) Given any orthonormal basis {ei }i∈I for H, there exists an orthonormal
basis {hi }i∈I for H such that
X
fi =
hωj , hi iej .
j∈I
(ii) {ωi }i∈I is an R-dual of {fi }i∈I .
(DTU Mathematics)
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Abstract duality
Corollary: (C., Kim, Kim) Assume that {ωj }j∈I is a tight orthogonal Riesz
sequence spanning a proper subspace of H with (Riesz) bound A and that
{fi }i∈I is a tight frame for H with frame bound A. Then the following hold:
(i) Given any orthonormal basis {ei }i∈I for H, there exists an orthonormal
basis {hi }i∈I for H such that
X
fi =
hωj , hi iej .
j∈I
(ii) {ωi }i∈I is an R-dual of {fi }i∈I .
As a special case we obtain the following known result:
Corollary: (Casazza, Kutyniok, Lammers) If {Emb Tna g}m,n∈Z is a tight frame
then { √1ab Em/a Tn/b g}m,n∈Z can be realized as the R-dual of {Emb Tna g}m,n∈Z .
(DTU Mathematics)
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Abstract duality
In order to gain further insight in the problem we will now consider the
following weaker version of Question B:
Question D: Let {fi }i∈I be a frame for H and {ωj }j∈I a Riesz sequence in H.
Under what conditions can we find an orthonormal basis {ei }i∈I for H and an
orthonormal sequence {hi }i∈I such that
fi =
X
hωj , hi iej ?
j∈I
(DTU Mathematics)
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Abstract duality
Theorem: Let {ωj }j∈I be a Riesz sequence in H having infinite deficit, and let
{ei }i∈I be an orthonormal basis for H. Then the following hold:
(i) For any sequence {fi }i∈I in H there exists an orthogonal sequence
{hi }i∈I in H such that
X
fi =
hωj , hi iej , ∀i ∈ I.
(8)
j∈I
(ii) Assume that {fi }i∈I is a Bessel sequence with bound B and that {ωj }j∈I
has a lower Riesz basis bound C ≥ B. Then there exists an orthonormal
sequence {hi }i∈I such that (8) holds.
(iii) For any Bessel sequence {fi }i∈I and regardless of the lower Riesz bound
for {ωj }j∈I , there exists a constant α > 0 such that
X
fi =
hαωj , hi iej , ∀i ∈ I.
j∈I
(DTU Mathematics)
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