# On Discrete Time Wilson Systems

Authors: Teena Kohli; Suman Panwar; S. K. Kaushik
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we define the discrete time Wilson frame (DTW frame) forl2ℤ and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.


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## Body

## 1. Introduction

The idea of frame as a redundant peer of a basis was originated in 1952 by Duffin and Schaeffer [1]. It came to limelight only with the historic paper of Daubechies et al. [2]. A sequence of vectors ujj∈ℕ⊆ℋ is termed as a frame (or Hilbert frame) for a separable Hilbert space ℋ if there exist constants Al,Au>0 such that(1)Alu2≤∑j∈ℕu,uj2≤Auu2,for allu∈ℋ.The positive numbersAl and Au are termed as lower and upper frame bounds of the frame, respectively. The bounds may not be unique. If Al=Au, then ujj∈ℕ is called an Al-tight frame, and if Al=Au=1, then ujj∈ℕ is said to be a Parseval frame. The inequality in (1) is recognized as the frame inequality of the frame ujj∈ℕ.A sequence of vectorsujj∈ℕ⊆ℋ is called a Riesz basis if ujj∈ℕ is complete and there are positive constants Al and Au such that(2)Al∑j∈ℕαj2≤∑j∈ℕαjuj2≤Au∑j∈ℕαj2,for allαjj∈ℕ∈ℓ2ℕ.Gabor frame forL2ℝ (which is a Riesz basis) has bad localization properties in either time or frequency. Thus, a system to replace Gabor systems which do not have bad localization properties in time and frequency was required. Wilson [3, 4] 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. [5] to construct orthonormal “Wilson bases” which consist of functions given by(3)ψjkx=εkcos2kπxwx−j2,ifjis even,2sin2k+1πxwx−j+12,ifjis odd,εk=2,ifk=0,2,ifk∈ℕ,with a smooth well-localized window function w. For such bases, the disadvantage described in the Balian–Low theorem is completely removed. Independent of the work of Daubechies et al. [5], orthonormal local trigonometric bases consisting of the functions wjcosk+1/2π−j,j∈ℤ,k∈ℕ0, where ℕ0=ℕ∪0, were introduced by Malvar [6], where window functions are assumed to be compactly supported, and only two immediately neighbouring windows are allowed to have overlapping support. Some generalizations of Malvar bases were studied in [7, 8]. To obtain more freedom for the choice of window functions, biorthogonal bases were investigated in [9]. A drawback of Malvar’s construction is the restriction on the support of the window functions. Therefore, it was preferred to consider Wilson bases of Daubechies et al. [5].Feichtinger et al. [10] proved that Wilson bases of exponential decay are not unconditional bases for all modulation spaces on ℝ including the classical Bessel potential spaces and the Schwartz spaces. Also, Wilson bases are not unconditional bases for the ordinary Lp spaces for p≠2, as shown in [10]. Approximation properties of Wilson bases are studied by Bittner [11], and Wilson bases for general time-frequency lattices are studied by Kutyniok and Strohmer [12]. Generalizations of Wilson bases to nonrectangular lattices are discussed by Sullivan et al. [3], with motivation from wireless communication and cosines modulated filter banks. Wojdyllo [13] studied modified Wilson bases and discussed Wilson system for triple redundancy in [14]. Discrete time Wilson frames with general lattices are studied by Lian et al. [15]. Motivated by the fact that one has different trigonometric functions for odd and even indices, Bittner [11, 16] considered Wilson bases introduced by Daubechies et.al [5] with nonsymmetrical window functions for odd and even indices. This generalized system of Bittner was later studied extensively by Kaushik and Panwar [17–19] and Jarrah and Panwar [20].In this article, we consider the system defined by Bittner [16] to define the discrete time Wilson frame (DTWF) and give examples for its existence. Some observations related to properties of discrete time Wilson frames are given. Also, a relationship between DTW frames and the discrete time Gabor frames is discussed. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are obtained and the frame operator for the DTW frame is constructed. Finally, dual pair of frames for discrete time Wilson systems is defined and a sufficient condition for its existence is given.The discrete time Wilson (DTW) system associated withg0,g−1∈l2ℤ is defined as(4)ψmM,kL=Em/MTkL/2+E−m/MTkL/2g0,ifk∈2ℤ,k≠0,1iEm+1/MTk+1L/2−E−m+1/MTk+1L/2g−1,ifk∈2ℤ+1,12Em/M+E−m/Mg0,ifk=0,where k∈ℤ, L,M∈ℕ and m=0,1,2,…,M−1.The DTW system given by (4) can be rewritten for any n∈ℤ as(5)ψm/M,kLn=2cos2πmnMg0n,ifk=0,2cos2πmnMg0n−kL2,ifk∈2ℤ,k≠0,2sin2πm+1nMg−1n−k+1L2,ifk∈2ℤ+1.Remark 1.
Forg0=g−1=g, the DTW system has the form(6)ψm/M,kLg=Em/MTkL/2+E−m/MTkL/2g,ifk∈2ℤ,k≠0,1iEm+1/MTk+1L/2−E−m+1/MTk+1L/2g,ifk∈2ℤ+1,12Em/M+E−m/Mg,ifk=0,where k∈ℤ, L,M∈ℕ and m=0,1,2,…,M−1.


## 2. Outline of the Paper

In this article, we define discrete time Wilson frames (DTW frames) and discuss various properties of DTW frames (see Observations (I) to (VIII)). An interplay between DTW frames and discrete time Gabor frames has been given in Theorem1. Also, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given in Theorem 3 and 4, respectively. The construction of the frame operator for the DTW frame is discussed in Theorem 5. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for its existence. Various examples are given to illustrate the discussion.


## 3. Discrete Time Wilson Frames

In this section, we define the discrete time Wilson frame based on the Wilson system considered by Bittner [11, 16], explore their existence through examples, and investigate various properties including its relationship with discrete time Gabor systems. We begin with the following definition.Definition 1.
The discrete time Wilson system:(7)ψm/M,kL:g0,g−1∈l2ℤ,k∈ℤ,L,M∈ℕ,m=0,1,2,…,M−1,where ψm/M,kL is as defined in (4) and is called a discrete time Wilson frame (DTWF) if there exist constants 0<Al≤Au<∞ satisfying(8)Alf2≤∑k∈ℤ∑m=0M−1f,ψm/M,kL2≤Auf2,for allf∈l2ℤ.
The constantsAl and Au are called lower and upper frame bounds for the DTWF ψm/M,kL:g0,g−1∈l2ℤ. The supremum of all lower frame bounds and the infimum of all upper frame bounds are called optimal lower and optimal upper frames bounds, respectively.
In case the systemψm/M,kL:g0,g−1∈l2ℤ,k∈ℤ,L,M∈ℕ,m=0,1,2,…,M−1 satisfy only the right-hand side of inequality (8), then the system is called a discrete time Wilson Bessel sequence for l2ℤ.
In order to show the existence of discrete time Wilson Bessel sequences which are not DTWF forl2ℤ, we give the following examples.Example 1.
(i)
Letgnn∈ℤ=en,n∈ℕ. Then,(9)∑k∈ℤ∑m=0M−1f,Em/MTkLg2=∑k∈ℤ∑m=0M−1fn+kL2=M∑k∈ℤfn+kL2≤Mf2,for allf∈l2ℤ.Therefore, we obtain(10)∑k∈ℤ∑m=0M−1f,ψm/M,kLg2≤4Mf2,for allf∈l2ℤ.Hence,ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a discrete time Bessel sequence for l2ℤ with Bessel bound 4M.However, it is a DTW frame if and only ifL=1.(ii)
Letgn=1/n,n=1,2,3,…,A,whereA<L<M,L≥2,0,otherwise.Then, we have(11)∑k∈ℤ∑m=0M−1f,Em/MTkLg2≤Mf2,for allf∈l2ℤ.Hence,ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a discrete time Bessel sequence for l2ℤ with Bessel bound 4M. Furthermore, it is not a frame as it does not satisfy the lower frame condition for fnn∈ℤ=eL∈l2ℤ.Moreover, note that(1)
IfA=L, then ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame with frame bounds A=2M and B=4M.(2)
IfA=L=M, then ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame with frame bounds A=M and B=4M.
Next, we give examples of Wilson systems which are discrete time Wilson frames forl2ℤ.Example 2.
(i)
Letgn=1/M,n=0,1,2,…,L−1,L<M,0,otherwise.Then, using the fact that∑m=0M−1e2πim/Mq−p=M,ifq−p∈Mℤ,0,ifq−p∉Mℤ,we have(12)∑k∈ℤ∑m=0M−1f,Em/MTkLg2=1M∑k∈ℤ∑m=0M−1∑p=0L−1fp+kL2=f2,for allf∈l2ℤ,∑m=0M−1f,cos2πm·Mg·2≤32f2,for allf∈l2ℤ.Therefore,ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a discrete time Wilson frame for l2ℤ.(ii)
Letgn=1/2M,n∈0,L∩ℤ,L<M,0,otherwise.Note that∑m=0M−1e2πim/Mq−p=M,ifq−p∈Mℤ,0,ifq−p∉Mℤ.Therefore,(13)∑k∈ℤ∑m=0M−1f,Em/MTkLg2=M22Mf2+∑k∈ℤfk+1L2,2∑m=0M−1f,cos2πm·Mg·2≤3M22Mf2,f∈l2ℤ.
Hence,ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ.
In view of the above discussion, we have the following observations in relation to DTW frames.(I)Letf,g∈l2ℤ and let TkL be the translation operator on l2ℤ, where k∈ℤ and L∈ℕ. Then,(14)∑k∈ℤ∑m=0M−1Imf,cos2πm.MTkLg.f¯,sin2πm.MTkLg.¯=0.Indeed, it follows from the fact that(15)∑m=0M−1e2πim/Mq−p=M,ifq−p∈Mℤ,0,ifq−p∉Mℤ.(II) Letψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be a DTW system for l2ℤ. Then,(16)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=4∑k∈ℤ∑m=0M−1f,cos2πm·MTkLg0·2+4∑k∈ℤ∑m=0M−1f,sin2πm·MTkLg−1·2−2∑m=0M−1f,cos2πm·Mg0·2,for allf∈l2ℤ.Indeed, one can compute that(17)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=4∑k∈2ℤk≠0∑m=0M−1f,cos2πm.MTkL/2g0·2+4∑k∈2ℤ+1k≠0∑m=0M−1f,sin2πm+1.MTk+1L/2g−1·2+2∑m=0M−1f,cos2πm.Mg0·2=4∑k∈ℤ∑m=0M−1f,cos2πm.MTkLg0·2−4∑m=0M−1f,cos2πm.Mg0·2+4∑k∈ℤ∑m=0M−1f,sin2πm.MTkLg−1·2+2∑m=0M−1f,cos2πm.Mg0·2.In view of Observations (I) and (II), we obtain (III).(III) Letψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be a DTW system for l2ℤ. Then,(18)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=4∑k∈ℤ∑m=0M−1f,Em/MTkLg0·2+4∑k∈ℤ∑m=0M−1f,Em/MTkLg−1·2−4∑k∈ℤ∑m=0M−1f,sin2πm.MTkLg0·2−2∑m=0M−1f,cos2πmMg0·2−4∑k∈ℤ∑m=0M−1f,cos2πm·MTkLg−1·2,f∈l2ℤ.Using Observations (II) and (III), we have (IV).(IV) For allf∈l2ℤ,(19)∑k∈ℤ∑m=0M−1f,Em/MTkLg0·2+∑k∈ℤ∑m=0M−1f,Em/MTkLg−1·2=∑k∈ℤ∑m=0M−1f,cos2πm·MTkLg0·2+∑k∈ℤ∑m=0M−1f,sin2πm·MTkLg0·2+∑k∈ℤ∑m=0M−1f,cos2πm·MTkLg−1·2+∑k∈ℤ∑m=0M−1f,sin2πm·MTkLg−1·2.(V) Ifg0=g−1=g for the DTW system ψm/M,kLgm=0,k∈ℤ,L,M∈ℕM−1, then for all f∈l2ℤ, and we obtain(20)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=4∑k∈ℤ∑m=0M−1f,Em/MTkLg.2−2∑m=0M−1f,cos2πmMg.2.(VI) LetEm/MTkLg0m=0,k∈ℤ,L,M∈ℕM−1 and Em/MTkLg−1m=0,k∈ℤ,L,M∈ℕM−1 be two DTG Bessel sequences with Bessel bounds B1 and B2, respectively. Then, the system ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence with Bessel bound 4B1+B2.Indeed, using observation (III) and the hypothesis, we have(21)∑k∈ℤ∑m=0M−1f,ψm/M,kL2≤4∑k∈ℤ∑m=0M−1f,Em/MTkLg0.2+4∑k∈ℤ∑m=0M−1f,Em/MTkLg−1.2≤4B1+B2‖f‖2,for allf∈l2ℤ.Remark 2.
The converse of observation (VI) may not be true even if additionally we assume that the systemψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a frame for l2ℤ.Example 3.
LetL=2,M=4, L=2,M=4, and g−1nn∈ℤ=e1. Then,(22)∑k∈ℤ∑m=0M−1f,Em/MTkLg02=4∑k∈ℤf2k2,for allf∈l2ℤ,∑k∈ℤ∑m=0M−1f,Em/MTkLg−12=4∑k∈ℤf2k+12,for allf∈l2ℤ.Thus, the systemsEm/MTkLg0m=0,k∈ℤ,L,M∈ℕM−1 and Em/MTkLg−1m=0,k∈ℤ,L,M∈ℕM−1 are not DTG frames for l2ℤ. Now, using observation (III), we obtain(23)8f2≤∑k∈ℤ∑m=0M−1f,ψm/M,kLg2≤16f2,for allf∈l2ℤ.Hence,ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ with frame bounds 8 and 16.(VII) Letg0,g−1∈l2ℤ be such that(24)B1=Msupn∈0,L∩ℕ∑p∈ℤ∑k∈ℤg0n−kLg0n−kL−pM¯<∞,B2=Msupn∈0,L∩ℕ∑p∈ℤ∑k∈ℤg−1n−kLg−1n−kL−pM¯<∞.Then, the systemψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence for l2ℤ with Bessel bound 4B1+B2.(VIII) Ifg0,g−1∈l2ℤ are functions having bounded support, then the DTW system ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence for l2ℤ.
Indeed, one may perceive that, since the functionsg0 and g−1 have bounded support, B1 and B2 as defined in observation (VII) are finite, and hence the DTW system ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence for l2ℤ.
Now, we prove a result related to DTW systems for the particular case wheng0=g−1=g.Lemma 1.
Forf,g∈l2ℤ, we have(25)2∑k∈ℤ∑m=0M−1f,Em/MTkLg2≤∑k∈ℤ∑m=0M−1f,ψm/M,kL2≤4∑k∈ℤ∑m=0M−1f,Em/MTkLg2.Proof.
Using observation (V), we obtain(26)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=4∑k∈ℤ∑m=0M−1f,Em/MTkLg⋅2−2∑m=0M−1f,cos2πm.Mg⋅2.
Hence, we compute(27)2∑m=0M−1f,cos2πm.Mg.2=2∑m=0M−1f,12Em/M+E−m/Mg.2=12∑m=0M−1f,Em/Mg+f,E−m/Mg2≤2∑m=0M−1f,Em/Mg2≤2∑k∈ℤ∑m=0M−1f,Em/MTkLg2.
In the following result, we give an interplay between the DTW frame and DTG frame forl2ℤ.Theorem 1.
The Wilson systemψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ if and only if Em/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 is a DTG frame for l2ℤ.Proof.
LetAl and Au be the positive constants such that(28)Alf2≤∑k∈ℤ∑m=0M−1f,ψmM,kL2≤Auf2,for allf∈l2ℤ.
Then, using Lemma1, it is easy to conclude that Em/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 is a DTG frame for l2ℤ with frame bounds Al/4 and Au/2.
Conversely, letEm/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 be a DTG frame for l2ℤ. Then, there exist positive constants ℬl and ℬu such that(29)ℬlf2≤∑k∈ℤ∑m=0M−1f,Em/MTkLg2≤ℬuf2,for allf∈l2ℤ.
Again, by utilizing Lemma1, we deduce that ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame with frame bounds 2ℬl and 4ℬu.
Now, we define discrete time tight Wilson frame forl2ℤ and investigate their relationship with discrete time Gabor frame for l2ℤ.Definition 2.
The discrete time Wilson systemψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 given by (4) is called a discrete time tight Wilson frame (DTTWF) if there exists a constant C∞≥0 such that(30)∑k∈ℤ∑m=0M−1f,ψm/M,kL2=C1f2,for allf∈l2ℤ.
IfC∞=1, then the frame ψm/M,kL:g0,g−1∈l2ℤ is called Discrete Time Parseval frame.
Next, we state two results whose proofs can be worked out using Lemma1.Proposition 1.
Letψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be a DTTW frame for l2ℤ with frame bound C∞. Then, Em/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 is a DTG frame with frame bounds C∞/4 and C∞/2.Proposition 2.
LetEm/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 be a DTTG frame for l2ℤ with frame bound C∞. Then, ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ with frame bounds 2C∞ and 4C∞.


## 4. Discrete Zak Transform and Discrete Time Wilson Frames

Various properties of the Zak transform (continuous version) were studied by Janssen [21,22] and the discrete version is discussed by Heil [23] who gave the following definition of discrete Zak transform.Definition 3 (see [23]).
The discrete Zak transform of a sequencef∈l2ℤ is given by(31)Zfn,x=∑j∈ℤfn+jae2πijx,∀n,x∈ℤ×ℝ^,where a∈ℤ+ is a fixed parameter and ℝ^ is the dual group of ℝ.
Next, we state a result related to Zak transform proved by Heil [23].Theorem 2 (see [23]).
Given a fixedg∈L2ℝ and L∈ℤ+. If L=M, then the system Em/MTkLg is a frame for l2ℤ with frame bounds D1 and D2⇔0<M−1D1≤Zg2≤M−1D2<∞a.e.
Now, we give a necessary condition for DTW frame in terms of the discrete Zak transform.Theorem 3.
LetL=M. If ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ with frame bounds Dl and Du, then(32)0<L−1Dl4≤Zg2≤L−1Du2<∞a.e.Proof.
Sinceψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ with frame bounds Dl and Du, using Theorem 1, the system Em/MTkLgm=0,k∈ℤ,L,M∈ℕM−1 is a DTG frame for l2ℤ with frame bounds Dl/4 and Du/2.
Hence, the result follows using Theorem2.
Towards, the converse of Theorem3, we have the following result.Theorem 4.
LetL=M. If there exists Dl>0 and Du>0 such that the following inequality holds(33)0<L−1Dl2≤Zg2≤L−1Du4<∞.a.e.,then ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW frame for l2ℤ with frame bounds Dl and Du.Proof.
It can be worked out using Theorem1 and Theorem 2.Remark 3.
ForL>M, the system ψm/M,kL:g∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is not a frame for l2ℤ.


## 5. Discrete Time Wilson Frame Operator

The frame operator for a frame is constructed by the composition of two important operators, namely, the analysis operator and the synthesis operator. The frame operator is positive, bounded, invertible, and self-adjoint. It ensures the existence of a canonical dual frame of a given frame, i.e., iffn is a frame and S is the frame operator, then S−1fn is a frame called the canonical dual of the frame fn. It is known that the canonical tight frame leads to a perfect reconstruction when used for both analysis and synthesis. Keeping this in mind, we make an attempt to construct the frame operator for the discrete time Wilson frame. We begin with the following definition.Definition 4.
Letψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be a discrete time Bessel sequence for l2ℤ. Then, DTWF operator S:l2ℤ⟶l2ℤ is defined as(34)Sf=∑k∈ℤ∑m=0M−1f,ψm/M,kLψm/M,kL,∀f∈l2ℤ.
In the following result, we construct the frame operator for the discrete time Wilson frame with the help of the frame operators of the two associated discrete time Gabor Bessel sequences.Theorem 5.
Forg0,g−1∈l2ℤ, let Em/MTkLg0m=0,k∈ℤ,L,M∈ℕM−1 and Em/MTkLg−1m=0,k∈ℤ,L,M∈ℕM−1 be DTG Bessel sequences with frame operators S1 and S2, respectively. Then, the frame operator S for DTW system ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is given by S=2S1+S2+P1−P2+R, where(35)S1f=∑k∈ℤ∑m=0M−1f,Em/MTkLg0Em/MTkLg0,S2f=∑k∈ℤ∑m=0M−1f,Em/MTkLg−1Em/MTkLg−1,P1f=∑k∈ℤ∑m=0M−1f,Em/MTkLg0E−m/MTkLg0,P2f=∑k∈ℤ∑m=0M−1f,Em/MTkLg−1E−m/MTkLg−1,Rf=∑m=0M−1f,Em/Mg0cos2πmM·g0·.Proof.
By hypothesis, we haveS1f=∑k∈ℤ∑m=0M−1f,Em/MTkLg0Em/MTkLg0 and S2f=∑k∈ℤ∑m=0M−1f,Em/MTkLg−1Em/MTkLg−1. Since Em/MTkLg0m=0,k∈ℤ,L,M∈ℕM−1 and Em/MTkLg−1m=0,k∈ℤ,L,M∈ℕM−1 are DTG Bessel sequences, we obtain(36)∑k∈ℤ∑m=0M−1f,Em/MTkLg2=∑k∈ℤ∑m=0M−1f,E−m/MTkLg2.
Also, using observation (VI), we deduce that the systemsE−m/MTkLg0m=0,k∈ℤ,L,M∈ℕM−1 and E−m/MTkLg−1m=0,k∈ℤ,L,M∈ℕM−1 are DTG Bessel sequences and the system ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence with their frame operators denoted by K1, K2, and S, respectively. Then, for all f∈l2ℤ, we obtain(37)K1f=∑k∈ℤ∑m=0M−1f,E−m/MTkLg0E−m/MTkLg0,K2f=∑k∈ℤ∑m=0M−1f,E−m/MTkLg−1E−m/MTkLg−1,Sf=∑k∈ℤ∑m=0M−1f,ψm/M,kLψm/M,kL=∑m=0M−1f,12Em/M+E−m/Mg012Em/M+E−m/Mg0+∑k∈2ℤ,k≠0∑m=0M−1f,Em/MTkL/2+E−m/MTkL/2g0Em/MTkL/2+E−m/MTkL/2g0+∑k∈2ℤ+1∑m=0M−1f,1iEm+1/MTk+1L/2−E−m+1/MTk+1L/2g−11iEm+1/MTk+1L/2−E−m+1/MTk+1L/2g−1=12∑m=0M−1f,Em/Mg0Em/Mg0+12∑m=0M−1f,E−m/Mg0Em/Mg0+12∑m=0M−1f,Em/Mg0E−m/Mg0+12∑m=0M−1f,E−m/Mg0E−m/Mg0+∑k∈2ℤ,k≠0∑m=0M−1f,Em/MTkL/2g0Em/MTkL/2g0+∑k∈2ℤ,k≠0∑m=0M−1f,E−m/MTkL/2g0Em/MTkL/2g0+∑k∈2ℤ,k≠0∑m=0M−1f,Em/MTkL/2g0E−m/MTkL/2g0+∑k∈2ℤ,k≠0∑m=0M−1f,E−m/MTkL/2g0E−m/MTkL/2g0+∑k∈2ℤ+1∑m=0M−1f,Em+1/MTk+1L/2g−1Em+1/MTk+1L/2g−1−∑k∈2ℤ+1∑m=0M−1f,E−m+1/MTk+1L/2g−1Em+1/MTk+1L/2g−1−∑k∈2ℤ+1∑m=0M−1f,Em+1/MTk+1L/2g−1E−m+1/MTk+1L/2g−1+∑k∈2ℤ+1∑m=0M−1f,E−m+1/MTk+1L/2g−1E−m+1/MTk+1L/2g−1nn,=S1f+S2f+K1f+K2f+P1f+T1f−P2f−T2f−R1f−R2f−R3f−R4f,where(38)P1f=∑k∈ℤ∑m=0M−1f,Em/MTkLg0E−m/MTkLg0,T1f=∑k∈ℤ∑m=0M−1f,E−m/MTkLg0Em/MTkLg0,P2f=∑k∈ℤ∑m=0M−1f,Em/MTkLg−1E−m/MTkLg−1,T2f=∑k∈ℤ∑m=0M−1f,E−m/MTkLg−1Em/MTkLg−1,R1f=12∑m=0M−1f,Em/Mg0Em/Mg0,R2f=12∑m=0M−1f,E−m/Mg0Em/Mg0,R3f=12∑m=0M−1f,Em/Mg0E−m/Mg0,R4f=12∑m=0M−1f,E−m/Mg0E−m/Mg0.
Now, for allh∈l2ℤ, we compute(39)T1f,h=∑k∈ℤ∑m=0M−1f,E−m/MTkLg0Em/MTkLg0,h=∑k∈ℤ∑m=0M−1∑p∈ℤ∑q∈ℤfpg0p−kL¯g0q−kLhq¯exp2πim/Mp+q=M∑k∈ℤ∑p,q∈ℤ,p+q∈Mℤfpg0p−kL¯g0q−kLhq¯,P1f,h=∑k∈ℤ∑m=0M−1f,Em/MTkLg0E−m/MTkLg0,h=∑k∈ℤ∑m=0M−1∑p∈ℤ∑q∈ℤfpg0p−kL¯g0q−kLhq¯exp−2πim/Mp+q=M∑k∈ℤ∑p,q∈ℤ,p+q∈Mℤfpg0p−kL¯g0q−kLhq¯.
Thus,T1f=P1f,for allf∈l2ℤ. Similarly, it can be proved that T2f=P2f,S1f=K1f,S2f=K2f,R1f=R4f,andR2f=R3f,for allf∈l2ℤ.
Hence, we conclude thatS=2S1+S2+P1−P2+R.


## 6. Dual Pair of Frames for Discrete Time Wilson Systems

In this section, we study dual pair of frames and obtain a sufficient condition for the existence of a dual pair of discrete time Wilson systems. First, we state the definition of a dual pair of frames discussed by Christensen [24, 25].Definition 5 (see [25]).
LetH be a Hilbert space and let fii∈I, gii∈I, pjj∈J, and qij∈J be Bessel sequences. Then, F=fii∈I∪pjj∈J and G=gii∈I∪qjj∈J are dual pair of frames if(40)f=∑i∈If,figi+∑j∈Jf,pjqj,for allf∈H.
In the following result, we give a sufficient condition for the existence of a dual pair of discrete time Wilson systems.Theorem 6.
LetL≤M and let ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and ξm/M,kL:w0,w−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be two DTW Bessel sequences for l2ℤ. Then, there exist DTW Bessel sequences Pm/M,kL:p0,p−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and Qm/M,kL:q0,q−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 such that P=ψm/M,kL∪Pm/M,kL and Q=ξm/M,kL∪Qm/M,kL are dual pair of frames for l2ℤ.Proof.
LetT and U be the preframe operators for the DTW Bessel sequences ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and ξm/M,kL:w0,w−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1, respectively. Then, T:l2ℤ×ℤM⟶l2ℤ and U:l2ℤ×ℤM⟶l2ℤ are given by(41)Tcm,k=∑k∈ℤ∑m=0M−1cm,kψm/M,kL,Ucm,k=∑k∈ℤ∑m=0M−1cm,kξm/M,kL,where ℤM=0,1,2,…,M−1. Then,(42)UT∗f=∑k∈ℤ∑m=0M−1f,ψm/M,kLξm/M,kL,for allf∈l2ℤ.
Also, the operatorΦ=I−UT∗ is bounded on l2ℤ. Furthermore, Φ∗=I−TU∗. Let R=Rm/M,kL:r0,r−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and S=Sm/M,kL:s0,s−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be a pair of DTW dual frames for l2ℤ. Then, using Proposition 2.1 of [25], we compute(43)f=∑k∈ℤ∑m=0M−1f,ψm/M,kLξm/M,kL+∑k∈ℤ∑m=0M−1f,Φ∗Rm/M,kLSm/M,kL.
Also, using Lemma 6.3.2 of [24], we deduce that R=ψm/M,kL∪Φ∗Rm/M,kL and S=ξm/M,kL∪Sm/M,kL form a dual pair of frames for l2ℤ if Φ∗Rm/M,kL is a DTW Bessel sequence. Now, observe that Φ∗Rm/M,kL is a DTW system given by Φ∗Rm/M,kL=Rm/M,kL∗=Rm/M,kL:Φ∗r0,Φ∗r−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1. Since R=Rm/M,kL:r0,r−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 is a DTW Bessel sequence and Φ is a bounded operator, Φ∗Rm/M,kL is a DTW Bessel sequence.
Finally, we prove a result related to compact support of functions generating DTW Bessel sequences.Theorem 7.
LetL≤M, and let ψm/M,kL:g0,g−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and ξm/M,kL:w0,w−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be two DTW Bessel sequences for l2ℤ. If the functions g0,g−1,w0, and w−1 are compactly supported, then the functions p0,p−1,q0, and q−1 are also compactly supported.Proof.
Suppose thatR=Rm/M,kL:r0,r−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 and S=Sm/M,kL:s0,s−1∈l2ℤm=0,k∈ℤ,L,M∈ℕM−1 be such that r0,r−1,s0, and s−1 be compactly supported. Then, in view of the proof of Theorem 6, one can conclude that the functions p0,p−1,q0, and q−1 are compactly supported if Φ∗r0 and Φ∗r−1 are compactly supported.
By assumption,g0,g−1,w0,w−1,r0,r−1,s0, and s−1 are all compactly supported. Therefore, there exists an N∈ℕ such that(44)g0n=g−1n=w0n=w−1n=s0n=s−1n=r0n=r−1n=0,for alln∉−N,N.
Since(45)Φ∗r0=I−TU∗r0=r0−∑k∈ℤ∑m=0M−1r0,ξm/M,kLψm/M,kL,Φ∗r0 is compactly supported. Similarly, Φ∗r−1 is compactly supported.


## 7. Conclusion

Gabor frame forL2ℝ (which is a Riesz basis) has bad localization properties in either time or frequency. Wilson [3, 4] suggested a system of functions which are localized around the positive and negative frequency of the same order. Based on the Wilson systems, Wilson frames for L2ℝ were introduced and studied in [17–20]. In this article, discrete time Wilson frames (DTWF) are defined and their relationship with discrete time Gabor frames is investigated. Also, frame operator for the DTWF has been constructed. Finally, keeping duality in mind, dual pair of frames for the discrete time Wilson systems have been studied.



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