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Abstract and Applied Analysis
Volume 2012, Article ID 746823, 14 pages
http://dx.doi.org/10.1155/2012/746823
Research Article

Characterizations of Irregular Multigenerator Gabor Frame on Periodic Subsets of ℝ

1Department of Mathematics, Shantou University, Guangdong, Shantou 515063, China
2Department of Mathematics, Hanshan Normal University, Guangdong, Chaozhou 521041, China

Received 10 March 2012; Accepted 19 May 2012

Academic Editor: Sergey V. Zelik

Copyright © 2012 D. H. Yuan 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.

Abstract

We consider the multigenerator system , for and , , where the parameters are not necessary the same. With the help of frame theory, we provide some sufficient or necessary conditions for the system to be a frame for . Moreover, we present some characterizations for this system to be a Parseval frame.

1. Introduction

For , consider the translation operator and the modulation operator , both acting on . We say that the system is a Gabor frame for if there exist two constants such that holds for every .

Gabor analysis is a pervasive signal processing method for decomposing and reconstructing signals from their time-frequency (TF) projections, and Gabor representation is used in many applications ranging from speech processing and texture segmentation to pattern and object recognition, among others. However, as it is widely recognized, a single-windowed Gabor expansion is not enough to analyze the dynamic TF contents of signals that contain a wide range of spatial and frequency components, the resolution of which is normally very poor. Therefore, if one could incorporate a set of multiple windows of various TF localizations in a frame system, the representation of signals of multiple and/or time-varying frequencies would have their corresponding windowing templates and resolutions to relate to. To this purpose, one of the best choices may be the multigenerator Gabor system.

Multigenerator Gabor system is firstly presented by Zibulski and Zeevi in [1]. Utilizing the piecewise Zak transform (PZT), they [2] discussed the frame operator associated with the multigenerator Gabor frame. They pointed out that the so-called Balian-Low theorem for multigenerator Gabor frame is generalized to consideration of a scheme of multigenerator which makes it possible to overcome in a way the constraint imposed by the original theorem in the case of a single window. Since then, researchers are interested in the study of both theory and application aspects of multigenerator Gabor frame; for detail, see [27]. Multigenerator Gabor systems may be both interesting and useful since they can increase the degree of freedom by incorporating windows of various types and widths.

Note that -periodic set in can be used to model a signal that appears periodically but intermittently. Recently, some authors concerned Gabor analysis in , where is an -periodic set in . Although classical Gabor analysis tools in can be adjusted to treat such a scenario by padding with zeros outside the set , Gabor systems that fit exactly such a scenario might have been more efficient. Gabardo and Li [8] obtained density results for Gabor systems associated with periodic subsets of the real line. Lian and Li [9] studied the Gabor frame sets for subspaces. They pointed out that only periodic in is suitable for Gabor analysis.

Motivated by [79], we address the issue about the multigenerator Gabor frame in this paper. With the help of frame theory, we provide some sufficient or necessary conditions for the multigenerator Gabor frame system to be a frame for , and we obtain the characterization for the multigenerator Gabor system to be a Parseval frame.

2. Notations

In this section, we present some notations and lemmas, which will be needed in the rest of the paper. Let be -periodic subset of . Then, is -periodic subset of for any given . Denote and where and . Given a measurable set in and a constant . Define. Consider the relation between , in : if and only if where denotes the cardinality of for a set . Then, it is easy to check that has an equivalence relation . Moreover, define Note that is an equivalence class under the relation or a partition of . Thus for and . Obviously, for given . It follows that for , and .

Example 2.1. Define for . Consider . Then However, it is easy to check that , which means .

Remark 2.2. Note that is the subset as defined in (2.2) of [9]. We point out that Proposition  2.1 (v) in [9] is incorrect.

Definition 2.3. Let for . We say that the system is a frame for if there exist two constants such that holds for every ; moreover, we say the frame for is tight if . In particular, the frame for is Parseval if .

Given a frame for , a dual frame of for is a frame for which satisfies the reconstruction property

For fixed positive integer , let . For given , we say that the system , is a multigenerator Gabor frame for if it is a frame for . Given a multigenerator Gabor frame for , a dual multigenerator Gabor frame for is a multigenerator Gabor frame for any .

The following lemma follows from general characterizations of shift-invariant frames, see [10, Corollary  1.6.2]. Alternatively, it can be proved similarly to [11, Theorem  8.4.4].

Lemma 2.4. Let , , and suppose that Then is a Bessel sequences with upper frame bound for . If also then is a frame for with bounds and .

3. Sufficient and Necessary Conditions

In this section, we provide some sufficient and necessary conditions for a class of the multigenerator Gabor frame system to be a frame for .

Firstly, we obtain the following theorem for the multigenerator Gabor system with the parameters and , which discloses the relationship between the Gabor system and its subspace .

Theorem 3.1. Let and . Then the following results hold.(I)If the Gabor system , is a frame for , then it is a frame for .(II)If the Gabor system , is a Bessel sequence for with upper bound , then it is a Bessel sequence for with the same upper bound.

Proof. The part (I) follows from the fact that .
Next, we prove the second part. Suppose that the Gabor system ,, is a Bessel sequence for . Then there exists a constant such that Observe that since , , for all . It follows that Therefore, for all . This implies that , is a Bessel sequence for with the same upper bound .

Moreover, we have the following sufficient condition for the multigenerator Gabor system with the parameters and .

Theorem 3.2. Let and . Moreover, suppose that Then is a Bessel sequence for with upper frame bound . If also then is a frame for with bounds and . That means is a multigenerator Gabor frame for .

Proof . Define then and are -periodic functions. Thus Define where and . Then, one obtains from (3.8) that respectively. Note that . By Lemma 2.4, one obtains the results.

Remark 3.3. Theorem 3.2 is similar to [11, Theorem  8.4.4]. Note that our result extends [11, Theorem  8.4.4] to the multigenerator and the periodic subset cases.

The following theorem gives necessary condition for the system to be a multigenerator Gabor frame for . It depends on the interplay among the function , the corresponding translation parameters , , , , and the subset .

Theorem 3.4. Let and . Assume that is a multigenerator Gabor frame for with bounds and . Then,

Proof. Firstly, note that is a -periodic subset of . Therefore, for all and . Thus
The rest part of the proof is by contradiction. Assume that the upper condition in (3.11) is violated on . Then there exists a measurable set with measure such that Similar to the discussion in [11, Proposition  8.3.2], we can assume that for small . Note that is a -periodic subset of . We can assume further that . Define Then, there exists such that and If not, that is for all , then This contradicts to . Therefore, we can also assume that is contained in an interval of length and that is a subset of .
Now consider the function and note that . Then for any , the function has support in . Since the functions constitute an orthonormal basis for for every interval of length for fixed , we have Thus, Therefore, This contradicts to the assumption that is an upper frame bound for. A similar proof shows that if the lower condition in (3.11) is violated, then cannot be a lower frame bound for .

4. Parseval Multigenerater Gabor Frame

In applications of frames, it is inconvenient that the frame decomposition, stated in [12, Theorem  5.1.7], requires inversion of the frame operator. As we have seen in the discussion of general frame theory, one way of avoiding the problem is to consider tight frames. We will characterize Parseval multigenerater Gabor frames in this section. Noting that , we obtain from [11, Lemma  8.4.3] or [12, Lemma  9.1.4] the following lemma, which will be used in the rest of the section.

Lemma 4.1. Let be a bounded measure function with compact support and . Then for given .

Theorem 4.2. Let and . Moreover, assume that is a tight frame for with . Then, Moreover, if and denote by , then hold a.e. in .

Proof. Define Consider Note that is a tight frame for with . Then Again, we obtain from Lemma 4.1 that for fixed and . Thus, for any . This implies that Note that and we obtain the desired result (4.2) and its special case  (4.3).
Next, we prove (4.4). For fixed , we obtain from Lemma 4.1 that Then, This, together with (4.3), follows that A change of variable shows that the contribution in the above sum arising from any value of is the complex conjugate of the contribution form the value . Therefore, where for . Now we divide three cases to draw the result.
Case  1. . Note that is a -periodic set. Then for all . Therefore, Thus,
Case  2. for fixed . Then for all . Therefore,
Case  3. and for fixed . Consider and let be any interval in of length at most . Denote by and by . If , then a.e. or a.e., thus Now consider . Define a function by Then, by (4.15), It follows that , a.e. on . Since is an arbitrary interval of length at most , we conclude that , a.e. in . A direct computation shows that Thus, we obtain the desired results.

To proceed further, we need use the following symbols. For , define Then there exists a unique nonnegative integer such that

Theorem 4.3. Let be the unique nonnegative integer satisfying (4.24). Assume that and satisfy a.e. in . Then is a tight frame for with .

Proof . For fixed , we obtain from Lemma 4.1 that Then, where This, together with (4.25), follows that Define Then, we obtain from (4.26) that This, together with (4.30), follows that Therefore, is a tight frame for with .

Note that when . We obtain from Theorems 4.2 and 4.3 that following Theorem 4.4, which is a necessary and sufficient condition for multigenerater Gabor Parseval frame.

Theorem 4.4. Let and ,. Then , is a tight frame for with if and only if hold a.e. in .

Remark 4.5. If , then is -periodic set for any given . In this case, , is a tight frame for with if and only if hold a.e. in .

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11071152) and the Natural Science Foundation of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511); this research was also partially supported by the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (Grant no. 201206012).

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