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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 747268, 8 pages
Regularity of Dual Gabor Windows
1Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
2Department of Mathematical Sciences, KAIST, 373-1 Guseong-Dong, Yuseong-Gu, Daejeon 305-701, Republic of Korea
3Department of Mathematics, Yeungnam University, 214-1 Dae-Dong, Gyeongsan-Si, Gyeongsangbuk-Do 712-749, Republic of Korea
Received 29 May 2013; Accepted 25 August 2013
Academic Editor: Anna Mercaldo
Copyright © 2013 Ole Christensen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 of there is a smooth function which simultaneously is a dual window for all B-spline generated Gabor frames for B-splines of order with a fixed and sufficiently small value of .
A frame in a separable Hilbert space leads to expansions of arbitrary elements , in a similar fashion as the well-known orthonormal bases. More precisely, there exists at least one so-called dual frame, that is, a framesuch that Unlessis a basis, the dualis 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 , that is, frames for that for a certain fixed parameter and a fixed function have the form . The function is called the window function. We will construct dual frames of the form for a suitable function to be called the dual window.
It is known that a frame will be overcomplete if , with the redundancy increasing when 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 there is a smooth function which simultaneously is a dual window for all B-spline generated Gabor frames for B-splines of orderwith a fixed and sufficiently small value of.
Just to give the reader an impression of the results to come, consider the B-spline . The function is continuous but not differentiable at the points . The Gabor system is a frame for all sufficiently small values of . As tends to zero, the redundancy of the frame increases, meaning that we get more and more freedom in the choice of a dual window . However, we will show that it is impossible to find a differentiable dual window supported on , regardless of the considered . On the other hand, by a seemingly innocent scaling we obtain the function . Again, is a frame for all sufficiently small values of . But in contrast with the situation for , we will show that one can find infinitely often differentiable dual windows that are supported on , for any value of . 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 Section 2. In Section 3, we consider a particular case where it is possible to obtain smooth dual windows, regardless of the regularity of the given window. 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 windows that are supported in . 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  and Kim . 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 , which in the special case of Gabor frames also lead to a class of dual Gabor frames; later, in , 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  and Janssen , respectively.
Lemma 1. Two Bessel sequencesand form dual frames forif and only if
We will use the following to apply Lemma 1.
Lemma 2. Letbe a real-valued bounded function, and assume that for some constant, Then, there exists a real-valued bounded function with such that
Proof. ConsiderWe will defineandsimultaneously. In caseputandOn the other hand, if , we know that . In this case, put and . Clearly, we can take outside .
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 function with support on an interval , we will in the rest of the paper use the function Due to the periodicity of, we are mainly interested in. Note that by the compact support of,
Theorem 3. Let , and let . Letbe a real-valued bounded function with, for which
for a constant. Then, the following hold. (i)The functionin (5) satisfies the conditions in Lemma 2. (ii)Takeas in Lemma 2, and let
Then,andform dual frames forand is supported in.
Proof. To prove (i), we note that by (7) and the definition of, for, Thus,satisfies the condition (3) in Lemma 2. Also, it is clear thatis bounded, real-valued, and -periodic. Therefore, we can choose a functionwhich is supported in and such that Defineas in (8). In order to prove (ii), we will apply Lemma 1. By assumption, the function has compact support and is bounded; by the construction, the function shares these properties. It follows thatand are Bessel sequences. In order to verify that these sequences form dual frames, we need to check that for, By assumption and construction,andhave support in; thus, (11) is satisfied forwhenever, that is, if. For, and using the compact support of, we need to check that For each, if, then Thus, we have This together with (6) and (10) implies, for, that Hence, (12) holds. Therefore,and form dual frames for.
3. Smooth Dual Windows for a Class of Gabor Frames
Before we start the general analysis of the dual windows in Theorem 3, 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. Let , and let . Let be a real-valued bounded function with , for which Letbe any bounded function for which Define the functionby and let Then, the following holds. (i)is a symmetric function with.(ii)andare dual frames for.(iii)Ifis chosen to be smooth, then the functionin (18) is smooth, and consequently the dual windowin (19) is smooth as well.
Proof. By the assumption (16), we have Takeas in (18). For, for , That is,Combining this with (20) yields By Theorem 3, is a dual window ofBy construction,is a symmetric function with . The result in (iii) follows by direct investigation of the derivatives at.
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 Example 9.
Another interesting feature of the construction in Theorem 4 is that the dual windowin (19) is independent of the windowthat generates the frame In other words, we can construct a windowthat generates a dual frame for all the frames satisfying the conditions in Theorem 4 for fixed values ofand.
Corollary 5. Let , and let . Consider a bounded real-valued function that is supported on and satisfies the partition of unity condition, Then, the functiongenerates a Gabor frameChoosing as in (18), the function in (19) is a dual window of.
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 , are given inductively byAny B-spline satisfies the partition of unity condition, and the B-spline has support on the interval. Thus, for each fixed value of, the function in (19) is a dual window for each of the B-splines and a fixed choice of .
4. Regularity of the Dual Windows if
Based on Theorem 3, we now aim at a general analysis of the relationship between the regularity of a windowand the associated dual windowswith 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 that are supported in . 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 functionthat is supported in , let
Theorem 6. Letbe a real-valued bounded function with . Assume that for some constant, Then, the following assertions hold. (1) If is not differentiable at , then there does not exist a differentiable dual window with for any ; (2) Assume that is differentiable at . Assume further that the set is a finite union of intervals and points not containing , that the set is finite, and that (a); (b). Then, for any , there exists a differentiable dual window with.
Note that (28) is a necessary condition for 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 Then, , and , so ,. Hence,satisfies condition (b) but not (a) in Theorem 6. Now we will show thatdoes not have a differentiable dual of any form. Suppose that there exists such a dual. By the duality condition, we obtain Letting and denote the left, respectively, and right derivatives of, we note that Taking the left and right derivatives of (30) at , this implies that But by conditions (30) and (31), we have. This is a contradiction. Thus, a differentiable dual window does not exist.
Note that the conclusion in Example 7 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 Easy considerations as in Example 7 show that satisfies condition (a) but not (b) in Theorem 6 and that 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 Section 1.
Example 9. Consider the B-spline, which is continuous but not differentiable at the points . It is an easy consequence of the results in the literature (see, e.g., ) that the Gabor system is a frame for all sufficiently small values of . As tends to zero, the redundancy of the frame increases, meaning that we get more and more freedom in the choice of . However, since is nondifferentiable at , Theorem 6 implies that none of the dual windows supported on are differentiable, for any .
On the other hand, consider the scaled B-spline . Again, is a frame for all sufficiently small values of . The requirements in Theorem 4 are satisfied for , 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 Theorem 6. The main difference between the results is that the conditions in the general version are stated in terms of the function in (5) rather than itself. The proof is in the appendix. Given a compactly supported function , define as in (5), and let
Theorem 10. Let , and let . Let be a real-valued bounded function with . Define as in (5). Then, the following assertions hold.
(1) If is not differentiable at , then there does not exist a differentiable dual windowdefined as in (8).
Assume that, for some constant, and that the setis a finite union of intervals and points not containing.
(2) Assume that is differentiable at , that the set of points is finite, and that (a); (b).
Then, is a frame for , and there exists a differentiable dual window of the form (8).
Note that the conditions in Theorem 10 (2) are void if is differentiable, that is, the standing assumptions alone imply the existence of a differentiable dual window.
For nonnegative functions, the conditions in Theorem 10 can be formulated in an easier way, where we again refer directly to properties of the function rather than .
Corollary 11. Let , and let . Let be a nonnegative bounded function with. Assume that for some constant,
Assume that the set of zeros of on is a finite union of intervals and points, that is differentiable except on a finite set of points, and that the sets and satisfy the following conditions: (a); (b); (c).
Then, , is a frame for , and there exists a differentiable dual window supported on .
Proof. We check condition (2) in Theorem 10. Let By an argument similar to the one at the beginning of the proof of Theorem 3, (36) together with (6) implies that Sinceis nonnegative, the zeros ofrestricted toare A direct calculation shows that ifis differentiable except on the set, thenis differentiable outside the set. Let Then, the setsandsatisfy the conditions (a) and (b) of (2) in Theorem 10. Hence, there exists a differentiable dual window.
Example 12. Let A direct calculation shows that Thus, the conditions (a)–(c) in Corollary 11 are satisfied. Hence, for any , there exists a differentiable dual window.
Proof of Theorem 10
The full proof of Theorem 10 is notationally complicated. We will therefore formulate the proof for a function for which (i)in (6), that is,, ; (ii) is differentiable except at one point, that is, ; (iii)the zeroset of within consists of just one interval (containing the degenerate case of just one point as a special case); that is, .
We leave the obvious modifications to the general case to the reader.
We use the following abbreviation:
Proof of Theorem 10. (1) We will prove the contrapositive result, so suppose that a dual window defined as in (8) with, that is,
with , is differentiable on . Then, is also differentiable on. Then, we have
The duality condition can for be written as
or, in terms ofand,
It follows from (A.3) that
Putting this into (A.5), we have
or, by dividing with ,
This implies that
Using (A.3) it now follows that
The conditions (A.3) and (A.4) and the continuity of show that is continuous at . Hence, , which implies that is differentiable at .
(2) We construct a real-valued differentiable functionwithso that
Let There are various scenarios concerning the location of the points and . They are treated in a similar way, and we will assume that . By the condition (2) in Theorem 10, one of the following cases occurs: (1); (2). The cases are similar, so we only consider the case (1), that is, Let
We defineon as follows: first, we defineon by which is well defined by (A.14). Then,
We now extend on choose onso that is differentiable on and Then, (A.16) and (A.18)-(A.19) imply thatis differentiable onand that We then defineonby which is well defined since This implies that Recall that , and are differentiable on . By (A.21), the same is the case for . Since , it remains to show that is differentiable on and that . Note that is differentiable on . Thus, by (A.16) and (A.20), it suffices to show that We first show (A.24). From (A.16), we get Putting this into (A.23), we have, for , or by dividing with, This implies that Using (A.18), it follows that Since is differentiable at , we have . This together with (A.27) implies that This proves that (A.24) holds. The results in (A.25) and (A.26) can be shown in a similar way.
The authors would like to thank the reviewers for their useful comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2010-0007614).
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