#### Abstract

Wilson frames as a generalization of Wilson bases have been defined and studied. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions and for odd and even indices of are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.

#### 1. Introduction

In 1946, Gabor [1] proposed a decomposition of a signal in terms of elementary signals, which displays simultaneously the local time and frequency content of the signal, as opposed to the classical Fourier transform which displays only the global frequency content for the entire signal. On the basis of this development, in 1952, Duffin and Schaeffer [2] introduced frames for Hilbert spaces to study some deep problems in nonharmonic Fourier series. In fact, they abstracted the fundamental notion of Gabor for studying signal processing. Janssen [3] showed that while being complete in , the set suggested by Gabor is not a Riesz basis. This apparent failure of Gabor system was then rectified by resorting to the concept of frames. Since then, the theory of Gabor systems has been intimately related to the theory of frames, and many problems in frame theory find their origins in Gabor analysis. For example, the localized frames were first considered in the realm of Gabor frames [4–7]. Gabor frames have found wide applications in signal and image processing. In view of Balian-Low theorem [8], Gabor frame for (which is a Riesz basis) has bad localization properties in time or frequency. Thus, a system to replace Gabor systems which does not have bad localization properties in time and frequency was required. For more literature on Gabor frames one may refer to [8–12]. Wilson et al. [13, 14] suggested a system of functions which are localized around the positive and negative frequency of the same order. The idea of Wilson was used by Daubechies et al. [15] to construct orthonormal “Wilson bases” which consist of functions given by with a smooth well-localized window function . For such bases the disadvantage described in the Balian-Low theorem is completely removed.

Independently from the work of Daubechies, Jaffard, and Journe, orthonormal local trigonometric bases consisting of the functions , , were introduced by Malvar [16]. Some generalizations of Malvar bases exist in [17, 18]. A drawback of Malvar's construction is the restriction on the support of the window functions. But the restriction on orthonormal bases allows only a small class of window functions. In [19], it has been proved that Wilson bases of exponential decay are not unconditional bases for all modulation spaces on including the classical Bessel potential space and the Schwartz spaces. Also, Wilson bases are not unconditional bases for the ordinary spaces for , shown in [19]. Approximation properties of Wilson bases are studied in [20]. Wilson bases for general time-frequency lattices are studied in [21]. Generalizations of Wilson bases to nonrectangular lattices are discussed in [13] with motivation from wireless communication and cosines modulated filter banks. Modified Wilson bases are studied in [22]. Bittner [23] considered the Wilson bases introduced by Daubechies et al. with nonsymmetrical window functions for odd and even indices of .

In this paper, we generalize the concept of Wilson bases and define Wilson frames. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions for odd and even indices of are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.

#### 2. Preliminaries

We assume that the reader is familiar with the theory of Gabor frames, refer [8, 9] for further details.

*Definition 1. *Let denote a Hilbert space. Let denote a countable index set. A family of vectors is called a frame for if there exist constants and with such that
The positive constants and are called lower frame bound and upper frame bound for the family , respectively. The inequality (2) is called the frame inequality. If in (2) only the upper inequality holds, then is called a Bessel sequence.

*Definition 2 (see [9]). *Let and are positive constants. The sequence is called a Gabor system for . Further,(i)if is a frame for , then it is called a Gabor frame; (ii)if is a Bessel sequence for , then it is called a Gabor Bessel sequence.

*Definition 3 (see [23]). *The Wilson system associated with is defined as a sequence of functions in given by
where and

If , then the Wilson system is given as .

*Definition 4 (see [9]). *The Zak transform of is defined as a function of two variables given by .

#### 3. Main Results

We begin this section with the definition of a Wilson frame.

*Definition 5. *The Wilson System for associated with is called a *Wilson frame* if there exist constants and with such that
The constants and are called lower frame bound and upper frame bound, respectively, for the Wilson frame .

*Definition 6. *In (5), if only the upper inequality holds for all , then the Wilson system , associated with , is called a Wilson Bessel sequence with Bessel bound .

*Example 7. *(a) Let . Then is a Wilson frame for .

(b) Let such that , for some constant and . Let . Consider the matrix

Let and . If , then the Wilson system is a Wilson frame for with bounds and .

(c) If we choose , then is not a Wilson frame for .

(d) Let
Then is a tight Wilson frame for with frame bound 2.

(e) Let . Then is a Wilson frame for .

(f) Let . Then is a Wilson frame for .

(g) If and , where , then is a Wilson frame for .

(h) Let

Then is a Wilson frame for .

Next, we give two Lemmas which will be used in the subsequent results. Lemma 8 is also proved in [24], but for the sake of completeness, we give the proof.

Lemma 8. *Let be the Wilson system associated with . Then, for ,
*

*Proof. *Let . Then
This gives
Thus,

Lemma 9. *Let be the Wilson system associated with . Then, for ,
*

*Proof. *We have
Therefore, using
we finally get the result.

*Remark 10. *Combining Lemmas 8 and 9, we get

*Remark 11. *In view of Lemma 8 and Remark 10, the Wilson system obtained by interchanging and is also a Wilson Bessel sequence if both the Gabor systems and are Bessel sequences.

The following result gives a necessary condition for a Wilson system associated with to be a Wilson frame.

The following result is motivated by Proposition in [9].

Theorem 12. *Let be a Wilson frame for associated with . Let denote its lower frame bound. Then
**
More precisely, if the inequality (17) is not satisfied, then the given Wilson system does not satisfy the lower frame condition.*

*Proof. *Assume that condition (17) is violated. Then there exists a measurable set having positive measure such that
We can assume that this is contained in an interval of length 1. Let
Then is partitioned into disjoint measurable sets such that at least one of these measurable sets will have a positive measure. Let this set be . Choose . Then , where = measure of .

Since for , the functions and have support in , constitute an orthonormal basis for , for every interval of length 1 and is contained in an interval of length 1, we have
Also, since , we have
Similarly, we obtain
Therefore,
Further, since
we get
Hence,
This is a contradiction.

Next, we give a sufficient condition for a Wilson system to be a Wilson Bessel sequence.

Theorem 13. *Let ,
**
Then, the Wilson system is a Wilson Bessel sequence with Bessel bound . *

*Proof. *In view of Theorem in [9], the Gabor systems and are Gabor Bessel sequences. Therefore, using Lemma 8 and Remark 10, the Wilson system is a Wilson Bessel sequence with Bessel bound .

Corollary 14. *Let be bounded and compactly supported. Then the Wilson system is a Wilson Bessel sequence. *

*Proof. *Since, are bounded and compactly supported, and as defined in Theorem 13 are both finite, and hence, the Wilson system is a Wilson Bessel sequence.

In the following results, we give a sufficient condition for the Wilson system to be a Wilson Bessel sequence in terms of Zak transforms of and .

Theorem 15. *Let , , and let there exist , such that and . Then, the Wilson system is a Wilson Bessel sequence with Bessel bound . *

*Proof. *In view of Proposition in [9], the Gabor systems and are Gabor Bessel sequences with Bessel bounds and , respectively. Therefore, using Lemma 8 and Remark 10, the Wilson system is a Wilson Bessel sequence with Bessel bound .

Next, we give sufficient conditions for a Wilson system to be a Wilson frame.

Theorem 16. *Let and
**
Then the Wilson system is a Wilson Bessel sequence. Further, if
**
then the Wilson system is a Wilson frame with frame bounds and . *

*Proof. *In view of Theorem in [9], the Gabor system is a Gabor frame for . If we choose in Lemma 9, then
Hence, the Wilson system is a Wilson frame with frame bounds and .

Corollary 17. *Suppose has support in an interval of length ; then the Wilson system is a Wilson frame for . *

*Proof. *Since has support in an interval of length , we have , for all . Thus, , . Hence, in view of Theorem 16, the Wilson system is a frame for .

The following result gives a class of functions for which the associated Wilson system is a Bessel sequence but not a frame.

Theorem 18. *Suppose is a continuous function with compact support. Then the Wilson system is a Wilson Bessel sequence for but not a frame.*

*Proof. *Since is a bounded function with compact support, defined in Theorem 16 is finite, and hence, the Wilson system is a Wilson Bessel sequence for . Moreover, since is a continuous function with compact support, in view of corollary in [9], the Gabor system can never become a frame, and hence, the Wilson system is not a Wilson frame.

Finally, we give a necessary and sufficient condition for a Wilson system to be a Wilson frame in terms of the Zak transform of .

Theorem 19. *The Wilson system is a Wilson frame for with bounds and if and only if . *

*Proof. *If we choose in Lemma 9, then
Therefore, the Wilson system is a Wilson frame for with bounds and if and only if the Gabor system is a frame for with bounds and . Since, the Gabor system is a frame for with bounds and if and only if , the result follows.

#### Acknowledgments

The authors thank the referee(s) for their useful suggestions and comments towards the improvement of the paper. The research of Suman Panwar is supported by CSIR vide letter no. 09045(1140)2011-EMR I dated 16112011.