An Interplay between Gabor and Wilson Frames

Abstract

Wilson frames {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} 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 𝑤₀ and 𝑤₋₁ for odd and even indices of 𝑗 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 𝐿²(ℝ), 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 𝐿²(ℝ) (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 𝜓^𝑘_𝑗(𝑥)=𝜀_𝑘()𝑤𝑗cos2𝑘𝜋𝑥𝑥−2(2())𝑤,if𝑗iseven,2sin𝑘+1𝜋𝑥𝑥−𝑗+12𝜀,if𝑗isodd,_𝑘=√2,if𝑘=0,2,if𝑘∈ℕ, (1) with a smooth well-localized window function 𝑤. 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 𝑤_𝑗cos(𝑘+(1/2))𝜋(⋅−𝑗), 𝑗∈ℤ, 𝑘∈ℕ₀ 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 ℝ including the classical Bessel potential space and the Schwartz spaces. Also, Wilson bases are not unconditional bases for the ordinary 𝐿^𝑝 spaces for 𝑝≠2, 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 𝑗.

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 𝑗 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 ℍ denote a Hilbert space. Let 𝐼 denote a countable index set. A family of vectors {𝑓_𝑖}_𝑖∈𝐼 is called a frame for ℍ if there exist constants 𝐴 and 𝐵 with 0<𝐴≤𝐵<∞ such that 𝐴𝑓²≤_𝑖∈𝐼𝑓,𝑓_𝑖²≤𝐵𝑓²,∀𝑓∈ℍ. (2) The positive constants 𝐴 and 𝐵 are called lower frame bound and upper frame bound for the family {𝑓_𝑖}_𝑖∈𝐼, respectively. The inequality (2) is called the frame inequality. If in (2) only the upper inequality holds, then {𝑓_𝑖}_𝑖∈𝐼 is called a Bessel sequence.

Definition 2 (see [9]). Let 𝑔∈𝐿²(ℝ) and 𝑎,𝑏 are positive constants. The sequence {𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔}_{𝑚,𝑛∈ℤ} is called a Gabor system for 𝐿²(ℝ). Further,(i)if {𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔}_{𝑚,𝑛∈ℤ} is a frame for 𝐿²(ℝ), then it is called a Gabor frame; (ii)if {𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔}_{𝑚,𝑛∈ℤ} is a Bessel sequence for 𝐿²(ℝ), then it is called a Gabor Bessel sequence.

Definition 3 (see [23]). The Wilson system associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ) is defined as a sequence of functions {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} in 𝐿²(ℝ) given by 𝜓^𝑘_𝑗(𝑥)=𝜀_𝑘cos(2𝑘𝜋𝑥)𝑤₀𝑗𝑥−2(2())𝑤,if𝑗iseven,2sin𝑘+1𝜋𝑥₋₁𝑥−𝑗+12,if𝑗isodd, (3) where ℕ₀=ℕ∪{0} and 𝜀_𝑘=√2,if𝑘=0,2,if𝑘∈ℕ. (4)

If 𝑤₀=𝑤₋₁=𝑔, then the Wilson system is given as {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0}.

Definition 4 (see [9]). The Zak transform of 𝑓∈𝐿²(ℝ) is defined as a function of two variables given by (𝑍𝑓)(𝑡,𝑣)=∑_𝑘∈ℤ𝑓(𝑡−𝑘)exp(2𝜋𝑖𝑘𝑣),𝑡,𝑣∈ℝ.

3. Main Results

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

Definition 5. The Wilson System {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} for 𝐿²(ℝ) associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ) is called a Wilson frame if there exist constants 𝐴 and 𝐵 with 0<𝐴≤𝐵<∞ such that 𝐴𝑓²≤_{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²≤𝐵𝑓²,∀𝑓∈𝐿²(ℝ). (5) The constants 𝐴 and 𝐵 are called lower frame bound and upper frame bound, respectively, for the Wilson frame {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0}.

Definition 6. In (5), if only the upper inequality holds for all 𝑓∈𝐿²(ℝ), then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0}, associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ), is called a Wilson Bessel sequence with Bessel bound 𝐵.

Example 7. (a) Let 𝑔=𝑤₀=𝑤₋₁=𝜒_[0,1). Then {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).
(b) Let 𝑤₀≠𝑤₋₁ such that |𝑤₋₁(𝑥)|≤𝐶(1+|𝑥|)^−1−𝜖, |𝑤₀(𝑥)|≤𝐶(1+|𝑥|)^−1−𝜖 for some constant 𝐶 and 𝜖>0. Let 𝑄⁺=(0,1/2)×[−1/2,1/2]. Consider the matrix 𝑀(𝑥,𝜉)=𝑍𝑤₀(𝑥,𝜉)𝑍𝑤₀(−𝑥,𝜉)−𝑍𝑤₋₁(𝑥,𝜉)𝑍𝑤₋₁(−𝑥,𝜉). (6)
Let 𝐴₀=essinf_{(𝑥,𝜉)∈𝑄}⁺‖𝑀⁻¹(𝑥,𝜉)‖₂⁻² and 𝐵₀=esssup_{(𝑥,𝜉)∈𝑄}⁺‖𝑀(𝑥,𝜉)‖²₂. If 0<𝐴₀≤𝐵₀<∞, then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ) with bounds 𝐴₀ and 𝐵₀.
(c) If we choose 𝑔=𝑤₀=𝑤₋₁=𝜒_[0,1/2), then {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is not a Wilson frame for 𝐿²(ℝ).
(d) Let 𝑤₀(𝑥)=sin𝜋𝑥𝜋𝑥,if𝑥≠0,1,otherwise. (7) Then {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a tight Wilson frame for 𝐿²(ℝ) with frame bound 2.
(e) Let 𝑔(𝑥)=𝑤₀(𝑥)=𝑤₋₁(𝑥)=2^1/2𝑒^−𝑥𝜒_[0,∞)(𝑥). Then {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).
(f) Let 𝑔(𝑥)=𝑤₀(𝑥)=𝑤₋₁(𝑥)=2^1/2/(1+2𝜋𝑖𝑥). Then {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).
(g) If 𝑤₀(𝑥)=𝑒^{−𝜉(𝑥−1/4)2} and 𝑤₋₁(𝑥)=𝑒^{−𝜉(𝑥+1/4)2}, where 𝜉>0, then {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).
(h) Let 𝑔(𝑥)=[),𝑥1+𝑥,if𝑥∈0,12,if𝑥∈[1,2),0,otherwise. (8)
Then {𝜓^𝑘_𝑗∶𝑔∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).
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 {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} be the Wilson system associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ). Then, for 𝑓∈𝐿²(ℝ), _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)². (9)

Proof. Let 𝑓∈𝐿²(ℝ). Then _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=_{𝑗∶even𝑘∈ℕ0}𝑓(𝑥)𝜀_𝑘cos(2𝑘𝜋𝑥)𝑤₀𝑗𝑥−2𝑑𝑥²+_{𝑗∶odd𝑘∈ℕ0}2𝑓(𝑥)sin(2(𝑘+1)𝜋𝑥)𝑤₋₁𝑥−𝑗+12𝑑𝑥². (10) This gives _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_𝑗∈ℤ𝑓(𝑥)𝑤₀𝑥−𝑗𝑑𝑥²+4_{𝑗∈ℤ𝑘∈ℕ}𝑓(𝑥)()×cos2𝑘𝜋𝑥𝑤₀𝑥−𝑗𝑑𝑥²+4_{𝑗∈ℤ𝑘∈ℕ}(𝑥)()×𝑓sin2𝑘𝜋𝑥𝑤₋₁𝑥−𝑗𝑑𝑥². (11) Thus, _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_𝑗∈ℤ𝑓(𝑥)𝑇_𝑗𝑤₀(𝑥)𝑑𝑥²+4_{𝑗∈ℤ𝑘∈ℕ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+4_{𝑗∈ℤ𝑘∈ℕ}𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)²=2_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)². (12)

Lemma 9. Let {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} be the Wilson system associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ). Then, for 𝑓∈𝐿²(ℝ), _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²+_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁²+_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)²−_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)². (13)

Proof. We have _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=_{𝑗∶even𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²+_{𝑗∶odd𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_𝑗∈ℤ𝑓,𝑇_𝑗𝑤₀²+_{𝑗∈ℤ𝑘∈ℕ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀+𝐸_−𝑘𝑇_𝑗𝑤₀²+_{𝑗∈ℤ𝑘∈ℕ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁−𝐸_−𝑘𝑇_𝑗𝑤₋₁²=2_𝑗∈ℤ𝑓,𝑇_𝑗𝑤₀²+_{𝑗,𝑘∈ℤ𝑘≠0}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²+_{𝑗,𝑘∈ℤ𝑘≠0}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁²+2Re_{𝑗,𝑘∈ℤ𝑘≠0}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀𝑓,𝐸_−𝑘𝑇_𝑗𝑤₀−2Re_{𝑗,𝑘∈ℤ𝑘≠0}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁𝑓,𝐸_−𝑘𝑇_𝑗𝑤₋₁. (14) Therefore, using Re𝑓,𝐸_𝑘𝑇_𝑗𝑔𝑓,𝐸_−𝑘𝑇_𝑗𝑔=𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑔(⋅)²−𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑔(⋅)², (15) we finally get the result.

Remark 10. Combining Lemmas 8 and 9, we get _{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²+_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁²=_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+_{𝑗,𝑘∈ℤ}𝑓,cos(2𝑘𝜋⋅)𝑇_𝑗𝑤₀(⋅)²+𝑓,sin(2𝑘𝜋⋅)𝑇_𝑗𝑤₋₁(⋅)². (16)

Remark 11. In view of Lemma 8 and Remark 10, the Wilson system obtained by interchanging 𝑤₀ and 𝑤₋₁ is also a Wilson Bessel sequence if both the Gabor systems {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} and {𝐸_𝑘𝑇_𝑗𝑤₋₁}_{𝑘,𝑗∈ℤ} are Bessel sequences.

The following result gives a necessary condition for a Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁}_{𝑗∈ℤ,𝑘∈ℕ0} associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ) to be a Wilson frame.

The following result is motivated by Proposition 9.1.2 in [9].

Theorem 12. Let {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)_{𝑗∈ℤ,𝑘∈ℕ0} be a Wilson frame for 𝐿²(ℝ) associated with 𝑤₀,𝑤₋₁∈𝐿²(ℝ). Let 𝐴 denote its lower frame bound. Then 𝐴2≤_𝑗∈ℤ𝑤₀𝑥−𝑗²+𝑤₋₁𝑥−𝑗². (17) 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 Δ⊆ℝ having positive measure such that 𝑊(𝑥)=_𝑗∈ℤ𝑤₀𝑥−𝑗²+𝑤₋₁𝑥−𝑗²<𝐴2onΔ. (18) We can assume that this Δ is contained in an interval of length 1. Let Δ₀=(𝑥)≤𝐴𝑥∈Δ∶𝑊2,Δ−1_𝑘=𝐴𝑥∈Δ∶2−1𝑘(𝑥)<𝐴<𝑊2−1.𝑘+1 (19) Then Δ is partitioned into disjoint measurable sets such that at least one of these measurable sets will have a positive measure. Let this set be Δ_𝑘′. Choose 𝑓=𝜒_Δ𝑘′. Then ‖𝑓‖=|𝜒_Δ𝑘′|, where |𝜒_Δ𝑘′| = measure of 𝜒_Δ𝑘′.
Since for 𝑗∈ℤ, the functions 𝑓𝑇_𝑗𝑤₀ and 𝑓𝑇_𝑗𝑤₋₁ have support in Δ_𝑘′, {𝐸_𝑘}_𝑘∈ℤ constitute an orthonormal basis for 𝐿²(𝐼), for every interval 𝐼 of length 1 and Δ_𝑘′ is contained in an interval of length 1, we have _𝑘∈ℤ𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²=_𝑘∈ℤ𝑓𝑇_𝑗𝑤₀,𝐸_𝑘²=_ℝ𝑓(𝑥)²𝑤₀𝑥−𝑗²𝑑𝑥. (20) Also, since 𝑓=𝜒_Δ𝑘′, we have _{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²=_{𝑗∈ℤΔ𝑘′}𝑤₀𝑥−𝑗²𝑑𝑥. (21) Similarly, we obtain _{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁²=_{𝑗∈ℤΔ𝑘′}𝑤₋₁𝑥−𝑗²𝑑𝑥. (22) Therefore, _{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²+𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁²=_{𝑗∈ℤΔ𝑘′}𝑤₀𝑥−𝑗²+𝑤₋₁𝑥−𝑗²=𝑑𝑥_Δ𝑘′𝑊(𝑥)𝑑𝑥. (23) Further, since _{𝑗∈ℤ,𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²≤2_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²+𝑓,𝐸_𝑘𝑇_𝑗𝑤₋₁², (24) we get _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²≤2_Δ𝑘′𝑊(𝑥)𝐴𝑑𝑥≤22−1𝑘^′+1^Δ_𝑘′=2𝑑𝑥𝐴−𝑘^′𝑓+1². (25) Hence, _{𝑗,𝑘∈ℤ}𝑓,𝜓^𝑘_𝑗²<𝐴𝑓². (26) This is a contradiction.

Next, we give a sufficient condition for a Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} to be a Wilson Bessel sequence.

Theorem 13. Let 𝑤₀,𝑤₋₁∈𝐿²(ℝ), 𝐵₁=sup_{𝑥∈[0,1]𝑘∈ℤ𝑛∈ℤ}𝑤₀(𝑥−𝑛)𝑤₀(𝐵𝑥−𝑛−𝑘)<∞,₂=sup_{𝑥∈[0,1]𝑘∈ℤ𝑛∈ℤ}𝑤₋₁(𝑥−𝑛)𝑤₋₁(𝑥−𝑛−𝑘)<∞. (27) Then, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence with Bessel bound 2(𝐵₁+𝐵₂).

Proof. In view of Theorem 9.1.5 in [9], the Gabor systems {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} and {𝐸_𝑘𝑇_𝑗𝑤₋₁}_{𝑘,𝑗∈ℤ} are Gabor Bessel sequences. Therefore, using Lemma 8 and Remark 10, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence with Bessel bound 2(𝐵₁+𝐵₂).

Corollary 14. Let 𝑤₀,𝑤₋₁∈𝐿²(ℝ) be bounded and compactly supported. Then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence.

Proof. Since, 𝑤₀,𝑤₋₁∈𝐿²(ℝ) are bounded and compactly supported, 𝐵₁ and 𝐵₂ as defined in Theorem 13 are both finite, and hence, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence.

In the following results, we give a sufficient condition for the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} to be a Wilson Bessel sequence in terms of Zak transforms of 𝑤₀ and 𝑤₋₁.

Theorem 15. Let 𝑤₀, 𝑤₋₁∈𝐿²(ℝ), and let there exist 𝐵₁>0, 𝐵₂>0 such that |𝑍𝑤₀|²≤𝐵₁ and |𝑍𝑤₋₁|²≤𝐵₂. Then, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence with Bessel bound 2(𝐵₁+𝐵₂).

Proof. In view of Proposition 9.7.3 in [9], the Gabor systems {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} and {𝐸_𝑘𝑇_𝑗𝑤₋₁}_{𝑘,𝑗∈ℤ} are Gabor Bessel sequences with Bessel bounds 𝐵₁ and 𝐵₂, respectively. Therefore, using Lemma 8 and Remark 10, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀,𝑤₋₁∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence with Bessel bound 2(𝐵₁+𝐵₂).

Next, we give sufficient conditions for a Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} to be a Wilson frame.

Theorem 16. Let 𝑤₀∈𝐿²(ℝ) and 𝐵=sup_{[]𝑥∈0,1𝑘∈ℤ𝑛∈ℤ}𝑤₀(𝑥−𝑛)𝑤₀(𝑥−𝑛−𝑘)<∞. (28) Then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℤ0} is a Wilson Bessel sequence. Further, if 𝐴=inf_{𝑥∈[0,1]𝑛∈ℤ}𝑤₀(𝑥−𝑛)²−_{𝑘≠0𝑛∈ℤ}𝑤₀(𝑥−𝑛)𝑤₀(𝑥−𝑛−𝑘)>0, (29) then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame with frame bounds 𝐴/2 and 𝐵/2.

Proof. In view of Theorem 9.1.5 in [9], the Gabor system {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} is a Gabor frame for 𝐿²(ℝ). If we choose 𝑤₀=𝑤₋₁ in Lemma 9, then _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²∀𝑓∈𝐿²(ℝ). (30) Hence, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame with frame bounds 𝐴/2 and 𝐵/2.

Corollary 17. Suppose 𝑤₀∈𝐿²(ℝ) has support in an interval of length 1; then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ).

Proof. Since 𝑤₀∈𝐿²(ℝ) has support in an interval of length 1, we have ∑_𝑛∈ℤ𝑤₀(𝑥−𝑛)𝑤₀(𝑥−𝑛−𝑘)=0, for all 𝑘≠0. Thus, 𝐵<∞, 𝐴>0. Hence, in view of Theorem 16, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a frame for 𝐿²(ℝ).

The following result gives a class of functions 𝑤₀∈𝐿²(ℝ) for which the associated Wilson system is a Bessel sequence but not a frame.

Theorem 18. Suppose 𝑤₀∈𝐿²(ℝ) is a continuous function with compact support. Then the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ)}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence for 𝐿²(ℝ) but not a frame.

Proof. Since 𝑤₀∈𝐿²(ℝ) is a bounded function with compact support, 𝐵 defined in Theorem 16 is finite, and hence, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson Bessel sequence for 𝐿²(ℝ). Moreover, since 𝑤₀∈𝐿²(ℝ) is a continuous function with compact support, in view of corollary 9.7.4 in [9], the Gabor system {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} can never become a frame, and hence, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℕ0} is not a Wilson frame.

Finally, we give a necessary and sufficient condition for a Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℕ0} to be a Wilson frame in terms of the Zak transform of 𝑤₀.

Theorem 19. The Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ) with bounds 2𝐴 and 2𝐵 if and only if 𝐴≤|𝑍𝑤₀|²≤𝐵.

Proof. If we choose 𝑤₀=𝑤₋₁ in Lemma 9, then _{𝑗∈ℤ𝑘∈ℕ0}𝑓,𝜓^𝑘_𝑗²=2_{𝑗,𝑘∈ℤ}𝑓,𝐸_𝑘𝑇_𝑗𝑤₀²∀𝑓∈𝐿²(ℝ). (31) Therefore, the Wilson system {𝜓^𝑘_𝑗∶𝑤₀∈𝐿²(ℝ}_{𝑗∈ℤ,𝑘∈ℕ0} is a Wilson frame for 𝐿²(ℝ) with bounds 2𝐴 and 2𝐵 if and only if the Gabor system {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} is a frame for 𝐿²(ℝ) with bounds 𝐴 and 𝐵. Since, the Gabor system {𝐸_𝑘𝑇_𝑗𝑤₀}_{𝑘,𝑗∈ℤ} is a frame for 𝐿²(ℝ) with bounds 𝐴 and 𝐵 if and only if 𝐴≤|𝑍𝑤₀|²≤𝐵, 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. 09⧵045(1140)⧵2011-EMR I dated 16⧵11⧵2011.