On Discrete Time Wilson Systems

In this paper, we define the discrete time Wilson frame (DTW frame) for l2(Z) 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.


Introduction
e 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 u j j∈N ⊆H is termed as a frame (or Hilbert frame) for a separable Hilbert space H if there exist constants A l , A u > 0 such that (1) e positive numbers A l and A u are termed as lower and upper frame bounds of the frame, respectively. e bounds may not be unique. If A l � A u , then u j j∈N is called an A l -tight frame, and if A l � A u � 1, then u j j∈N is said to be a Parseval frame. e inequality in (1) is recognized as the frame inequality of the frame u j j∈N .
A sequence of vectors u j j∈N ⊆H is called a Riesz basis if u j j∈N is complete and there are positive constants A l and A u such that for all α j j∈N ∈ ℓ 2 (N). (2) Gabor frame for L 2 (R) (which is a Riesz basis) has bad localization properties in either time or frequency. us, 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. e idea of Wilson was used by Daubechies et al. [5] to construct orthonormal "Wilson bases" which consist of functions given by 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 w j cos((k + (1/2))π (− j)), j ∈ Z, k ∈ N 0 , where N 0 � 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. erefore, 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 L p 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. is generalized system of Bittner was later studied extensively by Kaushik and Panwar [17][18][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. e discrete time Wilson (DTW) system associated with g 0 , g − 1 ∈ l 2 (Z) is defined as 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 eorem 1. Also, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given in eorem 3 and 4, respectively. e construction of the frame operator for the DTW frame is discussed in eorem 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.

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.
e discrete time Wilson system: where ψ (m/M),kL is as defined in (4) and is called a discrete time Wilson frame (DTWF) if there exist constants for all f ∈ l 2 (Z).
e constants A l and A u are called lower and upper frame bounds for the DTWF ψ (m/M),kL : g 0 , g − 1 ∈ l 2 (Z) . e 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 system ψ (m/M),kL : g 0 , g − 1 ∈ l 2 (Z), 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 l 2 (Z).
In order to show the existence of discrete time Wilson Bessel sequences which are not DTWF for l 2 (Z), we give the following examples.
erefore, we obtain Hence, ψ (m/M),kL : g ∈ l 2 (Z) M− 1 m�0,k ∈ Z,L,M ∈ N is a discrete time Bessel sequence for l 2 (Z) with Bessel bound 4M. Furthermore, it is not a frame as it does not satisfy the lower frame condition for f(n) n∈Z � e L ∈ l 2 (Z). Moreover, note that is a DTW frame with frame bounds A � 2M and B � 4M.

Journal of Mathematics 3 (2) If
Next, we give examples of Wilson systems which are discrete time Wilson frames for l 2 (Z).
otherwise. en, using the fact that Hence, ψ (m/M),kL : g ∈ l 2 (Z) In view of the above discussion, we have the following observations in relation to DTW frames.
(I)Let f, g ∈ l 2 (Z) and let T kL be the translation operator on l 2 (Z), where k ∈ Z and L ∈ N. en, Indeed, it follows from the fact that , for all f ∈ l 2 (Z).
Indeed, one can compute that In view of Observations (I) and (II), we obtain (III).
Journal of Mathematics 5 then for all f ∈ l 2 (Z), and we obtain ,k ∈ Z,L,M ∈ N be two DTG Bessel sequences with Bessel bounds B 1 and B 2 , respectively. en, Indeed, using observation (III) and the hypothesis, we have Remark 2. e converse of observation (VI) may not be true even if additionally we assume that the system  Indeed, one may perceive that, since the functions g 0 and g − 1 have bounded support, B 1 and B 2 as defined in observation (VII) are finite, and hence the DTW system ψ (m/M),kL : g 0 , g − 1 ∈ l 2 (Z) Now, we prove a result related to DTW systems for the particular case when g 0 � g − 1 � g.
Proof. Using observation (V), we obtain Hence, we compute In the following result, we give an interplay between the DTW frame and DTG frame for l 2 (Z). Proof. Let A l and A u be the positive constants such that for all f ∈ l 2 (Z).
Next, we state two results whose proofs can be worked out using Lemma 1.

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]). e discrete Zak transform of a sequence f ∈ l 2 (Z) is given by 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 fixed g ∈ L 2 (R) and L ∈ Z + . If L � M, then the system E (m/M) T kL g is a frame for l 2 (Z) with frame bounds D 1 and Now, we give a necessary condition for DTW frame in terms of the discrete Zak transform.

Discrete Time Wilson Frame Operator
e frame operator for a frame is constructed by the composition of two important operators, namely, the analysis operator and the synthesis operator. e frame operator is positive, bounded, invertible, and self-adjoint. It ensures the existence of a canonical dual frame of a given frame, i.e., if f n is a frame and S is the frame operator, then S − 1 f n is a frame called the canonical dual of the frame f n . 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. 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.

Journal of Mathematics
where 〈f, E (m/M) T kL g 0 〉E − (m/M) T kL g 0 , Now, for all h ∈ l 2 (Z), we compute , us, T 1 f � P 1 f, for all f ∈ l 2 (Z). Similarly, it can be proved that Hence, we conclude that S � 2(S 1 + S 2 + P 1 − P 2 + R).

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]). Let H be a Hilbert space and let f i i∈I , g i i∈I , p j j∈J , and q i j∈J be Bessel sequences. en, F � f i i∈I ∪ p j j∈J and G � g i i∈I ∪ q j j∈J are dual pair of In the following result, we give a sufficient condition for the existence of a dual pair of discrete time Wilson systems. ξ (m/M),kL : w 0 , w − 1 ∈ l 2 (Z)} M− 1 m�0,k ∈ Z,L,M ∈ N be two DTW Bessel sequences for l 2 (Z). If the functions g 0 , g − 1 , w 0 , and w − 1 are compactly supported, then the functions p 0 , p − 1 , q 0 , and q − 1 are also compactly supported.

Conclusion
Gabor frame for L 2 (R) (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 L 2 (R) were introduced and studied in [17][18][19][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.

Data Availability
No data were used to support the findings of the study.