On a lower bound for a periodic uncertainty
constant
Elena A. Lebedeva
Mathematics and Mechanics Faculty, St. Petersburg State University
Universitetsky prosp., 28, Peterhof, St. Petersburg, 198504, Russia
St. Petersburg State Polytechnical University,
Polytechnicheskay 29, St. Petersburg, 195251, Russia
Email: ealebedeva2004@gmail.com
Abstract—An inequality refining the lower bound for a periodic
(Breitenberger) uncertainty constant is proved for a wide class
of functions. A formula connecting uncertainty constants for
periodic and non-periodic functions is extended to this class.
I. I NTRODUCTION
The Breitenberger uncertainty constant (UC) is commonly
used as a measure of localization for periodic functions.
It was introduced in 1985 by Breitenberger in [3]. It can
be derived from a general operator ”position-momentum”
approach as it is discussed in [4]. The Breitenberger UC has
a deep connection with the classical Heisenberg UC, which
characterizes localization of functions on the real line. There
exists a universal lower bound for both UCs (the uncertainty
principle). It equals 1/2 (see chosen normalization in Sec.
II). It is well known that the least value is attained on the
Gaussian function in the real line case and there is no such
function in the periodic case. At the same time, in [2] Battle
proves a number of inequalities specifying the lower bound
of the Heisenberg UC for wavelets. In particular, it is proved
that if a wavelet ψ 0 ∈ L2 (R) has a zero frequency centre
c0 ) := R ξ|ψ
c0 (ξ)|2 dξ/(R |ψ
c0 (ξ)|2 dξ) = 0, then the
c(ψ
R
R
Heisenberg UC is greater or equal to 3/2 (see [2, Theorem
1.4]).
The main contribution of this paper is an inequality refining
the lower bound of the Breitenberger UC for a wide class of
sequences of periodic functions (Theorem 5). This result is
somewhat analogous to Battle’s result mentioned above. Given
a sequence of periodic functions ψj , j ∈ Z+ , the conditions
|(ψj0 , ψj )| ≤ Ckψj k2 and limj→∞ qj ψbj (k)/kψj k = 0 (see (9)
and (7) in Theorem 5) correspond to a zero frequency centre
c0 ) = 0 and the wavelet admissibility condition ψ
c0 (0) = 0
c(ψ
respectively. The rest of restrictions (8), (10)–(12) in Theorem
5 mean some “regularity” of the sequence ψj . In [18], the following formula connecting UCs for periodic (U CB ) and nonperiodic (U CH ) functions is obtained limj→∞ U CB (ψjp ) =
P
U CH (ψ0 ), where ψjp (x) := 2j/2 n∈Z ψ 0 (2j (x + 2πn)),
j ∈ Z+ . In Step 3 of the proof of Theorem 5, we generalize
this formula and suggest a new proof of this fact. In Remark
1 – Remark 3, we discuss which classes of periodic wavelet
sequences satisfy the conditions of Theorem 5.
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
While there are sufficiently many results specifying the
lower and the upper bounds of the Heisenberg UC [1], [2],
[4], [5], [7], [9], [10], [11], [12], [15] and the upper bound
of the Breitenberger UC [13], [14], [17], [19], [20], to our
knowledge, there are not actually any results concerting to an
estimation of the lower bound for the Breitenberger UC in the
literature.
This work also has the following motivation. In [13], a
family of periodic Parseval wavelet frames is constructed. The
family has optimal time-frequency localization (the Breitenberger UC tends to 1/2) with respect to a family parameter,
and it has the best currently known localization (the Breitenberger UC tends to 3/2) with respect to a multiresolution
analysis parameter. In [13], the conjecture was formulated: the
Breitenberger UC is greater than 3/2 for any periodic wavelet
sequence (ψj )j∈Z+ such that (ψj0 , ψj )L2,2π = 0. Theorem
5 of this paper proves the conjecture for a wide class of
sequences of periodic functions under a milder restriction
(ψj0 , ψj )L2,2π ≤ Ckψj k2L2,2π . So the family constructed in
[13] has optimal localization with respect to both parameters
within the class of functions considered in Theorem 5.
II. N OTATIONS AND AUXILIARY RESULTS
Let L2,2π be the space of all 2π-periodic square-integrable
complex-valued functions,
with inner product (·, ·) given by
Rπ
(f, g) := (2π)−1 −π f (x)g(x) dx for any f, g ∈ L2,2π ,
p
and norm k · k := (·, ·). The Fourier series of a function
P
ikx
b
f ∈ L2,2π is defined by
k∈Z f (k)e R , where its Fourier
π
coefficient is defined by fb(k) = (2π)−1 −π f (x)e−ikx dx.
Let L2 (R) be the space of all square-integrable complexvalued functions,
with inner product (·, ·) given by (f, g) :=
R
−1
(2π)
f
(x)g(x)
dx for any f, g ∈ L2 (R), and norm k·k :=
R
p
(·, ·). The Fourier transform
of a function f ∈ L2 (R) is
R
defined by fb(ξ) := (2π)−1 R f (x)e−iξx dx.
Let us recall the definitions of the UCs and the uncertainty
principles.
Definition 1 ([8]): The Heisenberg UC of f ∈ L2 (R)
is the functional U CH (f ) := ∆(f )∆(fb) such that
∆2 (f ) := kf k−2 k(· − c(f ))f k2 ,
c(f ) := kf k−2 (·f, f ),
where ∆(f ), ∆(fb), c(f ), and c(fb) are called time
variance, frequency variance, time centre,
and frequency centre respectively.
It is clear that the time variance can be rewritten as
∆2 (f ) =
(·f, f )2
k · f k2
−
kf k2
kf k4
Using elementary properties of the Fourier transform, we
rewrite the frequency variance as
∆2 (fb) =
kif 0 k2
(if 0 , f )2
−
2
kf k
kf k4
(See [18] Lemmas 1 and 2, where this trick is explained in
detail).
Theorem 1 ([8], [4]; the Heisenberg uncertainty principle):
Let f ∈ L2 (R), then U CH (f ) ≥ 1/2, and the equality is
attained iff f is the Gaussian function.
Theorem 2 ([2], p. 137; refinement of the Heisenberg
R
uncertainty principle): If f ∈ L2 (R), c(fb) = 0, and R f = 0,
then U CH (f ) ≥ 3/2.
P
ik·
Definition 2 ([3]): Let f =
∈ L2,2π . The
k∈Z ck e
first trigonometric moment is defined as
Z π
X
1
eix |f (x)|2 dx =
ck−1 ck .
τ (f ) :=
2π −π
k∈Z
The angular variance of the function f is defined by
P
2 2
kf k4
k∈Z |ck |
varA (f ) := P
− 1.
2 − 1 =
|τ (f )|2
k∈Z ck−1 c̄k
The frequency variance of the function f is defined
by
P
P
2
2
2 2
k∈Z k |ck |
k∈Z k|ck |
varF (f ) := P
− P
2
2
|ck |2
k∈Z |ck |
Theorem 3 ([3], [17]; the Breitenberger uncertainty principle): Let f ∈ L2,2π , f (x) 6= Ceikx , C ∈ R, k ∈ Z.
Then U CB (f ) > 1/2 and there is no function such that
U CB (f ) = 1/2.
Now, we recall the notion of a tight frame. Let H be a
separable Hilbert space. If there exists a constant A > 0
such
P∞ that for 2any f ∈ 2H the following equality holds
= Akf k , then the sequence (fn )n∈N is
n=1 |(f, fn )|
called a tight frame for H. In the case A = 1, a
tight frame is called a Parseval frame. In addition, if
kfn k = 1 for all n ∈ N, then a Parseval frame forms an
orthonormal basis.
We are especially interested to get a refinement of the
Breitenberger uncertainty principle for the case of periodic
wavelet sequences. For our purposes, it is sufficient to consider
wavelet systems with one wavelet generator. We recall the
basic notions. In the sequel, we use the following notation
fj,k (x) := fj (x − 2π2−j k) for a function fj ∈ L2,2π .
Consider
functions ϕ0 , ψj ∈ L2,2π , j ∈ Z+ . If the set
Ψ := ϕ0 , ψj,k : j ∈ Z+ , k = 0, . . . , 2j − 1 forms a tight
frame (or a basis) for L2,2π then Ψ is said to be a periodic
tight wavelet frame (or a periodic wavelet
basis) for L2,2π .
Theorem 4 ([6]; the unitary extension principle for a
periodic setting): Let ϕj ∈ L2,2π , j ∈ Z+ , be a sequence
of 2π-periodic functions such that
lim 2j/2 ϕ
bj (k) = 1,
=
0
ϕ
bj (k) = µj+1
bj+1 (k).
k ϕ
2
(f , f )
kf k
+
.
kf k2
kf k4
p
The quantity U CB ({ck }) := U CB (f ) := varA (f )varF (f )
is called the Breitenberger (periodic) UC.
We consider also two additional terms to characterize the
first trigonometric moment (In another form, they are introduced in [18, Lemma 3]). Namely, by definition, put
1X
A(f ) :=
|ck−1 − ck |2 ,
(1)
2
k∈Z
1X
B(f ) :=
(ck−1 − ck )(ck−1 + ck )
2
k∈Z
P
for f (x) = k∈Z ck eikx ∈ L2,2π . It is clear that
!
X
X
2
A(f ) =
|ck | − <
ck−1 ck = kf k2 − <(τ (f )),
k∈Z
k∈Z
(2)
!
B(f ) = i=
X
k∈Z
ck−1 ck
= i=(τ (f )).
(3)
Let µjk ∈ C, j ∈ Z+ , k ∈ Z, be a two-parameter sequence
such that µjk+2j = µjk , and
k∈Z
0 2
k ∈ Z.
j→∞
(4)
Let ψj , j ∈ Z+ , be a sequence of 2π-periodic functions
defined using Fourier coefficients
ψbj (k) = λj+1
bj+1 (k),
k ϕ
(5)
where λjk ∈ C, λjk+2j = λjk , and
µjk
λjk
µjk+2j−1
λjk+2j−1
!
µjk
µjk+2j−1
j
j
λk
λk+2j−1
!
=
2
0
0
2
.
(6)
Then Ψ := ϕ0 , ψj,k : j ∈ Z+ , k = 0, . . . , 2j − 1 forms a
Parseval wavelet frame for L2,2π .
The sequences (ϕj )j∈Z+ , (ψj )j∈Z+ , (µjk )k∈Z , and (λjk )k∈Z
are
called
a scaling sequence, a wavelet
sequence, a scaling mask and a wavelet
mask respectively.
A periodic wavelet system can be constructed starting with
a scaling mask. Namely, let νkj be a sequence given by νkj =
Q∞
j
r
b
νk+2
j . We define ξj (k) :=
r=j+1 νk . If the above infinite
products converge, then the scaling sequence, scaling mask,
wavelet mask, and wavelet sequence are defined respectively
as
√
ϕ
cj (k) := 2−j/2 ξbj (k),
µjk := 2νkj ,
−j
λj := e2πi2 k µj j−1 ,
ψbj (k) := λj+1 ϕ
bj+1 (k).
k
k
k+2
III. M AIN RESULT
In the following theorem we prove an inequality for the
Breitenberger UC for a wide class of sequences of periodic
functions.
Theorem 5: Let ψj ∈ L2,2π , j ∈ N be periodic functions
such that
lim qj ψbj (k)/kψj k = 0 for |k| ≤ M (C),
(7)
lim qj−2 A(ψj0 )/kψj k2 = 0,
(8)
|(ψj0 , ψj )| ≤ Ckψj k2 ,
(9)
qj−2 kψj0 k2 ≤ Ckψj k2 ,
qj2 A(ψj ) ≤ Ckψj k2 ,
qj |B(ψj )| ≤ Ckψj k2
(10)
j→∞
j→∞
(11)
(12)
Step 4. Let us consider auxiliary functions fj , j ∈ N defined
by
c
fbj := fc
∗j (· + c(f∗j )),
where c(fb) is the frequency centre of the function f (see
Definition 1). Then it is well-known (see [16, Exercise 1.5.1]
) that c(fbj ) = 0 and
U CH (f∗j ) = U CH (fj ),
lim U CB (ψj ) = lim U CB (ψ∗j ) = lim U CH (f∗j )
j→∞
lim qj ψbj (k) = 0 for |k| ≤ M (C),
j→∞
lim qj−2 A(ψj0 ) = 0,
j→∞
|(ψj0 , ψj )| ≤ C,
qj−2 kψj0 k2 ≤ C,
qj2 A(ψj )
qj |B(ψj )| ≤ C.
≤ C,
j→∞
IV. D ISCUSION
Analyzing the conditions of Theorem 5 one could ask
a natural question about classes of periodic sequences that
satisfy these conditions. Are these conditions restrictive or
mild? Let us make some illuminating remarks.
Remark 1 (Wavelet sequence generated by periodization):
Let ψ 0 ∈ L2 (R) be a wavelet function on the real line. It
is natural to require that ·ψ 0 , i(ψ 0 )0 ∈ L2 (R). Otherwise
U CH (ψ 0 ) will be infinite. Put
X
p
ψj,k
(x) := 2j/2
ψ 0 (2j (x + 2πn) + k).
(15)
The sequence
j ∈ Z+ , k = 0, . . . , 2j − 1 is
said to be a periodic wavelet set generated by
p
periodization. Set ψjp := ψj,0
, j = 0, 1, . . . . Put qj =
2j . We claim that for a wavelet sequence (ψjp )j∈Z+ conditions
(10)-(12) are fulfilled, and limj→∞ U CB (ψjp ) is finite. If
c0 (ξ) = o(√ξ) as ξ → 0, and ·(ψ 0 )0 ∈ L2 (R),
additionally ψ
then condition (7), and (8) are also fulfilled respectively.
Indeed, in [18], it is proved that the quantities
kψjp kL2,2π , 2−2j k(ψjp )0 k2L2,2π , 2−j ((ψjp )0 , ψjp )L2,2π ,
Step 3. Let us introduce auxiliary functions f∗j , j ∈ N such
that
−1/2
−1
d
fc
∗j (qj k) = ψ∗j (k),
−1
−1
fc
∗j is linear on any interval [qj (k − 1), qj k],
(16)
(17)
where k ∈ Z. Since ψ∗j ∈ L2,2π , it follows that f∗j ∈ L2 (R)
for j ∈ Z. It turns out that
U CB (ψ∗j ) − U CH (f∗j ) → 0 as j → ∞.
Theorem 5 is proved.
n∈Z
It follows from (7), (2), (11), and (12) that
qj
j→∞
p
(ψj,k
)j,k ,
Step 2. Let us consider auxiliary functions ψ∗j ∈ L2,2π such
that
cj (k), |k| > M (C),
ψ
d
ψ∗j (k) =
(14)
0,
|k| ≤ M (C).
U CB (ψ∗j ) − U CB (ψj ) → 0 as j → ∞.
j→∞
= lim U CH (fj ) ≥ 3/2.
j→∞
Sketch of Proof:
Step 1. The Breitenberger UC is a homogeneous functional
of degree zero, that is U C(αf ) = U C(f ) for α ∈ C \ {0}. So
in the sequel, we consider the functions ψj /kψj k instead of
ψj . However, to avoid the fussiness of notations we keep the
former name for the function ψj . Thus we consider a sequence
(ψj )j∈Z+ , where kψj k = 1, and conditions (7) – (12) take the
form
(19)
−1
We check that |c(fc
∗j )| ≤ qj M (C) as j > j0 for some j0 ∈
N. Furthermore, using conditions (14) and (16) we conclude
−1
−1
that fc
∗j (ξ) ≡ 0 for ξ ∈ [−qj M (C), qj M (C)]. Therefore,
−j
the inequality |c(fc
∗j )| ≤ 2 M (C) as j > j0 for some j0 ∈ N
b
c
c
provides fj (0) = f∗j (c(f∗j )) = 0 as j > j0 for some j0 ∈ N.
Step 5. Since fbj (0) = 0 as j > j0 for some j0 ∈ N and
c(fbj ) = 0 it follows from Theorem 2 that U CH (fj ) ≥ 3/2 as
j > j0 for some j0 ∈ N. It remains to note that using (15),
(18), and (19) we obtain
√
where M (C) := 2(C +C 2C/3+1/6), C > 0 is an absolute
constant, and qj → +∞ as j → +∞. If limj→∞ U CB (ψj )
exists, then
lim U CB (ψj ) ≥ 3/2.
(13)
j ∈ N.
(18)
2 · 22j A(ψjp ), and 2j B(ψjp )
tend to
kψ 0 kL2 (R) , k(ψ 0 )0 k2L2 (R) , ((ψ 0 )0 , ψ 0 )L2 (R) ,
k · ψ 0 k2L2 (R) , and i(·ψ 0 , ψ 0 )L2 (R)
respectively. Therefore, conditions (10) - (12) hold true. Then
in [18] it is deduced that limj→∞ U CB (ψjp ) = U CH (ψ 0 ), so
cp (k) = 2−j/2 ψ
c0 (2−j k)
limj→∞ U CB (ψjp ) is finite. Since ψ
j
√
c0 (ξ) = o( ξ) as ξ → 0, it follows that (7) is satisfied
and ψ
c0 (ξ) = o(√ξ) as ξ → 0 is not
for any k ∈ N. The condition ψ
restrictive. If this condition is not fulfilled, then U CH (ψ0 ) =
∞. Indeed, suppose U CH (ψ0 ) is finite. Then (1 + | · |)ψ, (1 +
| · |)ψb ∈ L2 (R), and ψ 0 is absolutely continuous (see [16,
Theorem 1.5.2]). Therefore, ψ 0 (x) = O(x−3/2−ε ), ε > 0
c0 (ξ + h) − ψ
c0 (ξ)| = O(h1/2+ε/2 ) as
as x → ∞. Then |ψ
√
c
0
ξ ∈ R. And the condition ψ (ξ) = o( ξ) as ξ → 0 follows
b
from the fact that ψ(0)
= 0. Finally, limj→∞ 2A((ψjp )0 ) =
0 0
0 0
k · (ψ ) kL2 (R) and ·(ψ ) ∈ L2 (R) yields (8).
It is clear that combining the result limj→∞ U CB (ψjp ) =
U CH (ψ 0 ) of [18] and Theorem 2 we immediately get the
inequality limj→∞ U CB (ψjp ) ≥ 3/2, and to do so we do not
need condition (8). However the above inequality is proved
here under the condition ((ψjp )0 , ψjp )L2,2π = 0, while only the
mild restriction ((ψjp )0 , ψjp )L2,2π ≤ Ckψjp k2L2,2π is required in
Theorem 5, cf. (9).
Remark 2 (Wavelet sequence generated by UEP): Let
(ψj )j∈Z+ be a wavelet sequence satisfying Theorem 4 and
c1 ≤ kψj k ≤ c2 , where c1 > 0, c2 are absolute constants. Put qj = 2j/2 . Then condition (7) is fulfilled for
cj (k) =
k ∈ Z. In fact, by (5) and (6), we conclude that ψ
j+1
2πi2−j−1 k j+1
[
e
µk+2j ϕ
j+1 (k) and it follows from (6) |µk+2j | ≤
√
√
cj (k)| ≤ 2|[
2. Therefore, |ψ
ϕj+1 (k)|. Using (3), we get (7)
for all k ∈ N. Next, if ψj is a trigonometric polynomial of
degree less than C1 2j/2 , where C1 is an absolute constant,
then condition (10) is also fulfilled. Indeed, it follows from
the Bernstein inequality that 2−j kψj0 k2 /kψj k2 ≤ C02 .
Remark 3 (Wavelet sequence constructed in [13]): Let
(ψja )j∈N , a > 1, be a family of periodic wavelet sequences
constructed in [13]. Then all conditions (7)–(12) are hold
true for qj = j 1/2 . Indeed, it is straightforward to see that
ca (k) j −1 2−j/2 for a fixed k ∈ Z, kψ a k2 j −1/2 2−j ,
ψ
j
j
A((ψja )0 ) j −1/2 2−j , k(ψja )0 k2 j 1/2 2−j , A(ψja ) j −3/2 2−j , |B(ψja )| sin(2π2−j−1 )j −1/2 2−j as j → ∞.
ACKNOWLEDGMENT
The work is supported by the RFBR, grant #15-01-05796,
by Saint Petersburg State University, grant #9.38.198.2015.
R EFERENCES
[1] R. Balan, An uncertainty inequality for wavelet sets, Appl. Comput.
Harmon. Anal. 5 (1998) 106–108.
[2] G. Battle, Heisenberg inequalities for wavelet states, Appl. Comput.
Harmon. Anal. 4 (1997) 119–146.
[3] E. Breitenberger, Uncertainty measures and uncertainty relations for angle
observables, Found. Phys. 15 (1985) 353–364.
[4] G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical
survey, Journal of Fourier analysis and applications, 3 (3) (1997) 207–
238.
[5] Ž. Gimbutas, A. Bastys, Daubechies compactly supported wavelets with
minimal Heisenberg boxes, Lit. Math. J. 35 (1995) 343–362.
[6] S.S. Goh, K.M. Teo, Extension principles for tight wavelet frames of
periodic functions, Appl. Comput. Harmon. Anal. 25 (2008) 168–186.
[7] T.N.T. Goodman, S.L. Lee, Asymptotic optimality in time-frequency localization of scaling functions and wavelets, in: Frontiers in Interpolation
and Application, N.K. Govil, H.N. Mhaskar, R.N. Mohapatra, Z. Nashed,
and J. Szadados (Eds.), (2007) 145–171.
[8] W. Heisenberg, The actual concept of quantum theoretical kinematics and
mechanics, Physikalische Z., 43, (1927) 172.
[9] E.A. Lebedeva, Exponentially decaying wavelets with uncertainty constants uniformly bounded with respect to the smoothness parameter,
Siberian Math. J, 49, 3 (2008) 457–473.
[10] E.A. Lebedeva, Minimization of the uncertainty constant of the family
of Meyer wavelets. Math. Notes, 81, 3-4 (2007) 489–95.
[11] E.A. Lebedeva, On the uncertainty principle for Meyer wavelet functions, Journal of Mathematical Sciences, 182, 5 (2012) 656–662.
[12] E.A. Lebedeva, Quasispline wavelets and uncertainty constants, Appl.
Comput. Harmon. Anal. 30 (2011) 214-230.
[13] E.A. Lebedeva, J. Prestin, Periodic wavelet frames and timefrequency
localization, Appl. Comput. Harmon. Anal. 37 2 (2014) 347–359.
[14] F.J. Narcowich, J.D. Ward, Wavelets associated with periodic basis
functions, Appl. Comput. Harmon. Anal. 3 (1996) 40–56.
[15] Novikov I. Ya. Modified Daubechies wavelets preserving localization
with growth of smoothness, East J. Approximation, 1, 3 (1995) 314–348.
[16] I.Ya. Novikov, V.Yu. Protasov, M.A. Skopina, Wavelet Theory, Translations of Mathematical Monographs 239, AMS, 2011.
[17] J. Prestin, E. Quak, Optimal functions for a periodic uncertainty principle
and multiresolution analysis, Proc. Edinb. Math. Soc., II. Ser. 42 (1999)
225–242.
[18] J. Prestin, E. Quak, H. Rauhut, K. Selig, On the connection of uncertainty principles for functions on the circle and on the real line, J. Fourier
Anal. Appl. 9 (2003) 387–409.
[19] H. Rauhut, Best time localized trigonometric polynomials and wavelets,
Adv. Comput. Math. 22 (2005) 1–20.
[20] K. Selig, Trigonometric wavelets and the uncertainty principle, in:
Approximation Theory, M. W. Müller, M. Felten, and D. H. Mache (Eds.),
Math. Research, Vol. 86, Akademie Verlag, Berlin (1995) 293–304.