# Regularity of Dual Gabor Windows

Authors: Ole Christensen; Hong Oh Kim; Rae Young Kim
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value ofK∈ℕ, there is a smooth function h which simultaneously is a dual window for all B-spline generated Gabor frames {EmbTnBN(x/2)}m,n∈ℕ for B-splines BN of order N=1,…,2K+1 with a fixed and sufficiently small value of b.


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## Body

## 1. Introduction

A frame{fk} in a separable Hilbert space ℋ leads to expansions of arbitrary elements f∈ℋ, in a similar fashion as the well-known orthonormal bases. More precisely, there exists at least one so-called dual frame, that is, a frame{hk}such that
(1)f=∑‍〈f,hk〉fk,∀f∈ℋ.
Unless{fk}is a basis, the dual{hk}is not unique. This makes it natural to search for duals with special prescribed properties. In this paper, we will consider Gabor frames with translation parameter a=1, that is, frames for L2(ℝ) that for a certain fixed parameter b>0 and a fixed function g∈L2(ℝ) have the form {EmbTng}m,n∈ℤ:={e2πimbxg(x-n)}m,n∈ℤ. The function g is called the window function. We will construct dual frames of the form {EmbTnh}m,n∈ℤ={e2πimbxh(x-n)}m,n∈ℤ for a suitable function h∈L2(ℝ) to be called the dual window.It is known that a frame{EmbTng}m,n∈ℤ will be overcomplete if b<1, with the redundancy increasing when b decreases. Thus, the dual window is not unique. We will investigate whether this freedom can be used to find dual windows with higher regularity than the canonical dual and comparable size of the support. We will present cases where this is possible and other cases where it is not. As a special case of our results, we show that there are certain classes of interesting frames that have the same dual window. For example, for any value of K∈ℕ there is a smooth function h which simultaneously is a dual window for all B-spline generated Gabor frames {EmbTnBN(x/2)}m,n∈ℕfor B-splines BNof orderN=1,…,2K+1with a fixed and sufficiently small value ofb.Just to give the reader an impression of the results to come, consider the B-splineB2. The function B2 is continuous but not differentiable at the points 0,±1. The Gabor system {EmbTnB2}m,n∈ℤ is a frame for all sufficiently small values of b>0. As b tends to zero, the redundancy of the frame {EmbTnB2}m,n∈ℤ increases, meaning that we get more and more freedom in the choice of a dual window h. However, we will show that it is impossible to find a differentiable dual window h supported on suppB2=[-1,1], regardless of the considered b>0. On the other hand, by a seemingly innocent scaling we obtain the function g(x):=B2(x/2). Again, {EmbTng}m,n∈ℤ is a frame for all sufficiently small values of b>0. But in contrast with the situation for B2, we will show that one can find infinitely often differentiable dual windows h that are supported on suppg=[-2,2], for any value of b∈]0,1/2]. These examples illustrate that the question of differentiability of the dual windows is a nontrivial issue. The examples will be derived as a consequence of more general results; see Example 9.The results will be based on a construction of dual windows, as presented in Section2. In Section 3, we consider a particular case where it is possible to obtain smooth dual windows, regardless of the regularity of the given windowg. This is much more than one can hope for in the general case. A general approach to the question of differentiability of the dual windows is given in Section 5. Since the necessary conditions are quite involved and not very intuitive, we first, in Section 4, state a version for the case of windowsg that are supported in [-1,1]. For this case, we can provide concrete examples demonstrating that the desired conclusions might fail if any of the constraints is removed.Note that a complementary approach to duality for Gabor frames that also deals with the issue of regularity is considered by Laugesen [1] and Kim [2]. For more information about the theory for Gabor analysis and its applications, see [3–5].


## 2. Construction of Dual Frames

In the literature, various characterizations of the pairs of dual Gabor frames are available. For general frames, Li gave a characterization in [6], which in the special case of Gabor frames also lead to a class of dual Gabor frames; later, in [7], it was shown that these duals actually characterize all duals. In order to start our analysis, we need the duality conditions for Gabor frames by Ron and Shen [8] and Janssen [9], respectively.Lemma 1.
Two Bessel sequences{EmbTnag}m,n∈ℤand {EmbTnah}m,n∈ℤ form dual frames forL2(ℝ)if and only if
(2)∑k∈ℤ‍g(x+nb+ka)¯h(x+ka)=bδn,0,a.e.x∈[0,a].We will use the following to apply Lemma1.Lemma 2.
LetG~be a real-valued bounded function, and assume that for some constantA>0,
(3)|G~(x-1)|+|G~(x)|≥A,x∈[0,1].
Then, there exists a real-valued bounded function H~ with suppH~⊆suppG~∩[-1,1] such that
(4)G~(x-1)H~(x-1)+G~(x)H~(x)=1,x∈[0,1].Proof.
Considerx∈[0,1].We will defineH~(x)andH~(x-1)simultaneously. In case|G~(x-1)|≥A/2,putH~(x)=0andH~(x-1)=1/G~(x-1).On the other hand, if |G~(x-1)|<A/2, we know that |G~(x)|≥A/2. In this case, put H~(x-1)=0 and H~(x)=1/G~(x). Clearly, we can take H~=0 outside [-1,1].We will now present a general result about the existence of frames with a dual window of a special form. Note that, in contrast with most results from the literature, we do not need to assume that the integer translates of the window function form a partition of unity.Associated to a functiong with support on an interval [-(2K+1),2K+1],K∈ℕ, we will in the rest of the paper use the function
(5)G~(x):=∑k∈ℤ‍g(x+2k),x∈ℝ.
Due to the periodicity ofG~, we are mainly interested inx∈[-1,1]. Note that by the compact support ofg,
(6)G~(x)=∑k=-KK‍g(x+2k),x∈[-1,1].Theorem 3.
LetK∈ℕ∪{0}, and let b∈]0,1/(4K+2)]. Letgbe a real-valued bounded function withsuppg⊆[-(2K+1),2K+1], for which
(7)|∑n∈ℤ‍g(x+n)|≥A,x∈[0,1],
for a constantA>0. Then, the following hold.   (i)
The functionG~in (5) satisfies the conditions in Lemma 2.(ii)
TakeH~as in Lemma 2, and let
(8)h(x)=b∑k=-KK‍T2kH~(x).
Then,{EmbTng}m,n∈ℤand{EmbTnh}m,n∈ℤform dual frames forL2(ℝ),andh is supported in[-(2K+1),2K+1].Proof.
To prove (i), we note that by (7) and the definition ofG~, forx∈[0,1],
(9)|G~(x-1)|+|G~(x)|=|∑k∈ℤ‍g(x-1+2k)|+|∑k∈ℤ‍g(x+2k)|≥|∑k∈ℤ‍g(x+k)|≥A.
Thus,G~satisfies the condition (3) in Lemma 2. Also, it is clear thatG~is bounded, real-valued, and 2-periodic. Therefore, we can choose a functionH~which is supported in [-1,1]and such that
(10)G~(x-1)H~(x-1)+G~(x)H~(x)=1,x∈[0,1].
Definehas in (8). In order to prove (ii), we will apply Lemma 1. By assumption, the function g has compact support and is bounded; by the construction, the function h shares these properties. It follows that{EmbTng}m,n∈ℤand{EmbTnh}m,n∈ℤ are Bessel sequences. In order to verify that these sequences form dual frames, we need to check that forx∈[0,1],
(11)∑k∈ℤ‍g(x+nb+k)¯h(x+k)=bδn,0,a.e.x∈[0,1].
By assumption and construction,gandhhave support in[-(2K+1),2K+1]; thus, (11) is satisfied forn≠0whenever1/b≥4K+2, that is, ifb∈]0,1/(4K+2)]. Forn=0, and using the compact support ofg, we need to check that
(12)∑k=-2K-12K‍g(x+k)h(x+k)=b,x∈[0,1].
For eachk∈{-K,-K+1,…,K}, ifx∈[0,1], then
(13)x-1+2k∈[2k-1,2k],x+2k∈[2k,2k+1];
Thus, we have
(14)h(x-1+2k)=bH~(x-1),h(x+2k)=bH~(x).
This together with (6) and (10) implies, forx∈[0,1], that
(15)∑k=-2K-12K‍g(x+k)h(x+k)=∑k=-KK‍g(x-1+2k)h(x-1+2k)+∑k=-KK‍g(x+2k)h(x+2k)=b∑k=-KK‍g(x-1+2k)H~(x-1)+b∑k=-KK‍g(x+2k)H~(x)=b(G~(x-1)H~(x-1)+G~(x)H~(x))=b.
Hence, (12) holds. Therefore,{EmbTng}m,n∈ℤand {EmbTnh}m,n∈ℤform dual frames forL2(ℝ).


## 3. Smooth Dual Windows for a Class of Gabor Frames

Before we start the general analysis of the dual windows in Theorem3, we will consider a particular case, where we can construct smooth compactly supported dual windows, regardless of the regularity of the window itself.Theorem 4.
LetK∈ℕ∪{0}, and let b∈]0,1/(4K+2)]. Let g be a real-valued bounded function with suppg⊆[-(2K+1),2K+1], for which
(16)∑n∈ℤ‍g(x+2n)=1,x∈[-1,1].
Letf:ℝ→ℝbe any bounded function for which
(17)f(x)=0,x≤0,f(x)=1,x≥1.
Define the functionH~by
(18)H~(x)={12f(2(x+1)),-1≤x<-12,1-12f(-2x),-12≤x<0,1-12f(2x),0≤x<12,12f(2(1-x)),12≤x<1,0,otherwise,
and let
(19)h(x):=b∑k=-KK‍T2kH~(x).
Then, the following holds.   (i)
his a symmetric function withsupph⊆[-(2K+1),2K+1].(ii)
{EmbTng}m,n∈ℤand{EmbTnh}m,n∈ℤare dual frames forL2(ℝ).(iii)
Iffis chosen to be smooth, then the functionH~in (18) is smooth, and consequently the dual windowhin (19) is smooth as well.Proof.
By the assumption (16), we have
(20)G~(x)=1,x∈[-1,1].
TakeH~as in (18). Forx∈]0,1/2[,
(21)H~(x-1)+H~(x)=f(2x)2+(1-f(2x)2)=1;
for x∈]1/2,1[,
(22)H~(x-1)+H~(x)=(1-f(-2(x-1))2)+f(2(1-x))2=1.
That is,H~(x-1)+H~(x)=1,x∈[0,1].Combining this with (20) yields
(23)G~(x-1)H~(x-1)+G~(x)H~(x)=H~(x-1)+H~(x)=1,x∈[0,1].
By Theorem 3, h is a dual window ofg.By construction,his a symmetric function with supph⊆[-(2K+1),2K+1]. The result in (iii) follows by direct investigation of the derivatives at-1,-1/2,0,1/2,1.An example of a smooth functionfsatisfying condition (17) is (see [10, page 36])
(24)f(x)={exp[-{exp[x(1-x)]-1}-1],0<x<1,0,x≤0,1,x≥1.As noted in the introduction, the possibility of constructing a smooth dual window is a significant improvement compared to the use of the canonical dual window, which might not even be continuous. We return to this point in Example9.Another interesting feature of the construction in Theorem4 is that the dual windowhin (19) is independent of the windowgthat generates the frame{EmbTng}m,n∈ℤ! In other words, we can construct a windowhthat generates a dual frame for all the frames {EmbTng}m,n∈ℤ satisfying the conditions in Theorem 4 for fixed values ofKandb.Corollary 5.
LetK∈ℕ, and let b∈]0,1/(4K+2)]. Consider a bounded real-valued function ϕ that is supported on [-K,K] and satisfies the partition of unity condition,
(25)∑n∈ℤ‍ϕ(x-n)=1,x∈ℝ.
Then, the functiong(x):=ϕ(x/2)generates a Gabor frame{EmbTng}m,n∈ℤ.ChoosingH~ as in (18), the functionh in (19) is a dual window ofg.Proof.
By the choice of the functionϕ,suppg⊆[-2K,2K]⊆[-2K-1,2K+1], and
(26)∑n∈ℤ‍g(x-2n)=∑n∈ℤ‍ϕ(x2-n)=1.
ChooseH~as in (18). By Theorem 4, the functionhdefined byh(x)=b∑k=-KK‍T2kH~(x)is a dual window ofg.It is known that the partition of unity condition (25) is satisfied for a large class of functions, for example, any scaling function for a multiresolution analysis. As a concrete example, recall that the centered B-splines BN, N∈ℕ are given inductively byB1=χ[-1/2,1/2],BN+1=BN*B1.Any B-spline satisfies the partition of unity condition, and the B-splineBN has support on the interval[-N/2,N/2]. Thus, for each fixed value ofK∈ℕ, the function h in (19) is a dual window for each of the B-splinesBN,N=1,…,2K and a fixed choice of b≤1/(4K+2).


## 4. Regularity of the Dual Windows ifsuppg⊆[-1,1]

Based on Theorem3, we now aim at a general analysis of the relationship between the regularity of a windowgand the associated dual windowshwith comparable support size. We will exhibit cases where the smoothness can be increased and other cases where these dual windows cannot have higher smoothness than the window itself. The general version of our result, to be stated in Theorem 10, is quite complicated, and the role of the conditions is not intuitively clear. Therefore, we will first present, in Theorem 6, the corresponding version for windows g that are supported in [-1,1]. An advantage of this approach is that for each of the requirements in Theorem 6 we can provide an example showing that the desired conclusion might fail if the condition is removed.Given a functiongthat is supported in [-1,1], let
(27)Z:={x∈[-1,1]:g(x)=0},E:={x∈[-1,1]:gisnotdifferentiableatx}.Theorem 6.
Letgbe a real-valued bounded function with suppg⊆[-1,1]. Assume that for some constantA>0,
(28)|g(x-1)|2+|g(x)|2≥A,x∈[0,1].
Then, the following assertions hold.  (1)
Ifg is not differentiable at 0, then there does not exist a differentiable dual window h with supph⊆[-1,1] for any b>0;(2)
Assume thatg is differentiable at 0. Assume further that the set Z is a finite union of intervals and points not containing 0, that the set E is finite, and that
(a)
E∩(E+1)=∅;(b)
Z∩(E±1)=∅.Then, for anyb∈]0,1/2], there exists a differentiable dual window h withsupph⊆suppg.Note that (28) is a necessary condition for {EmbTng}m,n∈ℤ being a frame; thus, it is not a restriction in this context. To support the conditions in Theorem 6, we will now provide a series of examples, where just one of these conditions breaks down and the conclusion in Theorem 6 fails. We first give an example where condition (a) is not satisfied.Example 7.
Consider the function(29)g(x):={2x+2,x∈[-1,-12];1,x∈[-12,12];-2x+2,x∈[12,1];0,x∉[-1,1].
Then, K=0,Z={±1}, and E={±1/2,±1}, so E∩(E+1)={1/2},Z∩(E±1)=∅. Hence,gsatisfies condition (b) but not (a) in Theorem 6. Now we will show thatgdoes not have a differentiable dual of any form. Suppose that there exists such a dualh. By the duality condition, we obtain
(30)g(x-1)h(x-1)+g(x)h(x)=b,x∈[0,1].
LettingD-g andD+g denote the left, respectively, and right derivatives ofg, we note that
(31)g(-12)=g(12)=1,D-g(-12)=2,D+g(-12)=0,D-g(12)=0,D+g(12)=-2.
Taking the left and right derivatives of (30) at x=1/2, this implies that
(32)Dh(-12)+Dh(12)=-2h(-12),Dh(-12)+Dh(12)=2h(12).
But by conditions (30) and (31), we haveh(-1/2)+h(1/2)=b. This is a contradiction. Thus, a differentiable dual window does not exist.Note that the conclusion in Example7 is even stronger than what we asked for: no dual window at all can be differentiable, regardless of its form and support size. In the next example condition, (b) in Theorem 6 is not satisfied, and the conclusion breaks down.Example 8.
Consider(33)g(x):={52x+32,x∈[-35,-15];1,x∈[-15,15];-52x+32,x∈[15,35];0,x∉[-35,35].
Easy considerations as in Example 7 show that g satisfies condition (a) but not (b) in Theorem 6 and that g does not have a differentiable dual of any form. We leave the details to the interested reader.Let us now provide the details for the example mentioned in Section1.Example 9.
Consider the B-splineB2, which is continuous but not differentiable at the points 0,±1. It is an easy consequence of the results in the literature (see, e.g., [11]) that the Gabor system {EmbTnB2}m,n∈ℤ is a frame for all sufficiently small values of b>0. As b tends to zero, the redundancy of the frame {EmbTnB2}m,n∈ℤ increases, meaning that we get more and more freedom in the choice of h. However, since B2 is nondifferentiable at x=0, Theorem 6 implies that none of the dual windows supported on [-1,1] are differentiable, for any b∈]0,1/2].
On the other hand, consider the scaled B-splineg(x):=B2(x/2). Again, {EmbTng}m,n∈ℤ is a frame for all sufficiently small values of b>0. The requirements in Theorem 4 are satisfied for b∈]0,1/2], implying that infinitely often differentiable dual windows exists.


## 5. Regularity of the Dual Windows in the General Case

We will now present the general version of Theorem6. The main difference between the results is that the conditions in the general version are stated in terms of the function G~ in (5) rather than g itself. The proof is in the appendix. Given a compactly supported function g:ℝ→ℂ, define G~ as in (5), and let
(34)Z~:={x∈[-1,1]:G~(x)=0},E~:={x∈[-1,1]:G~χ[-1,1]isnotdifferentiableatx}.Theorem 10.
LetK∈ℕ∪{0}, and let b∈]0,1/(4K+2)]. Let g be a real-valued bounded function with suppg⊆[-(2K+1),2K+1]. Define G~ as in (5). Then, the following assertions hold.
(1) IfG~ is not differentiable at 0, then there does not exist a differentiable dual windowhdefined as in (8).
Assume that, for some constantA>0,
(35)|G~(x-1)|+|G~(x)|≥A,x∈[0,1]
and that the setZ~is a finite union of intervals and points not containing0.
(2) Assume thatG~ is differentiable at 0, that the set of points E~ is finite, and that  (a)
E~∩(E~-1)=∅;(b)
G~(x)≠0,x∈(E~-1)∪(E~+1).
Then,{EmbTng}m,n∈ℤ is a frame for L2(ℝ), and there exists a differentiable dual windowh of the form (8).Note that the conditions in Theorem10 (2) are void if G~ is differentiable, that is, the standing assumptions alone imply the existence of a differentiable dual window.For nonnegative functions, the conditions in Theorem10 can be formulated in an easier way, where we again refer directly to properties of the function g rather than G~.Corollary 11.
LetK∈ℕ∪{0}, and let b∈]0,1/(4K+2)]. Let g be a nonnegative bounded function withsuppg⊆[-(2K+1),2K+1]. Assume that for some constantA>0,
(36)∑k=-2K-12K‍g(x+k)≥A,x∈[0,1].
Assume that the setZ of zeros of g on [-(2K+1),2K+1] is a finite union of intervals and points, that g is differentiable except on a finite set E of points, and that the sets E and Z satisfy the following conditions:  (a)
0∉(⋂k=-KK‍(Z-2k))∪(⋃k=-KK‍(E-2k));(b)
(⋃k=-KK‍(E-2k))∩(⋃k=-KK‍(E-2k+1))=∅;(c)
(⋂k=-KK‍(Z-2k))∩(⋃k=-KK‍(E-2k±1))=∅.
Then,{EmbTng}m,n∈ℤ, is a frame for L2(ℝ), and there exists a differentiable dual window h supported on [-(2K+1),2K+1].Proof.
We check condition (2) in Theorem10. Let
(37)G~(x):=∑k∈ℤ‍g(x+2k).
By an argument similar to the one at the beginning of the proof of Theorem 3, (36) together with (6) implies that
(38)|G~(x-1)|+|G~(x)|≥A,x∈[0,1].
Sincegis nonnegative, the zeros ofG~restricted to[-1,1]are
(39)Z~:=(⋂k=-KK‍(Z-2k))∩[-1,1].
A direct calculation shows that ifgis differentiable except on the setE, thenG~is differentiable outside the set⋃k=-KK‍(E-2k). Let
(40)E~:=(⋃k=-KK‍(E-2k))∩[-1,1].
Then, the setsE~andZ~satisfy the conditions (a) and (b) of (2) in Theorem 10. Hence, there exists a differentiable dual windowh.Example 12.
Let(41)g(x):=(52x+2)χ[-4/5,-2/5](x)+χ[-2/5,2/5](x)+(-52x+2)χ[2/5,4/5](x).
A direct calculation shows that
(42)E={±25,±45},E+1={15,35,75,95},E-1={-95,-75,-35,-15},Z=[-1,-45]∪[45,1].
Thus, the conditions (a)–(c) in Corollary 11 are satisfied. Hence, for any b∈]0,1/2], there exists a differentiable dual windowh.



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