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
(1)ψjk(x)={εkcos(2kπx)w(x-j2),if j is even,2sin(2(k+1)πx)w(x-j+12),if j is odd,εk={2,if k=0,2,if k∈ℕ,
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 wjcos(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 Lp 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.

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

Definition 5.
The Wilson System {ψjk: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
(5)A∥f∥2≤∑j∈ℤk∈ℕ0|〈f,ψjk〉|2≤B∥f∥2, ∀f∈L2(ℝ).
The constants A and B are called lower frame bound and upper frame bound, respectively, for the Wilson frame {ψjk: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 {ψjk: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 {ψjk: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
(6)M(x,ξ)=(Zw0(x,ξ)¯Zw0(-x,ξ)¯-Zw-1(x,ξ)¯Zw-1(-x,ξ)¯).

Let A0=ess inf(x,ξ)∈Q+∥M-1(x,ξ)∥2-2 and B0=ess sup(x,ξ)∈Q+∥M(x,ξ)∥22. If 0<A0≤B0<∞, then the Wilson system {ψjk: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 {ψjk:g∈L2(ℝ)}j∈ℤ,k∈ℕ0 is not a Wilson frame for L2(ℝ).

(d) Let
(7)w0(x)={sinπxπx,if x≠0,1,otherwise.
Then {ψjk: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)=21/2e-xχ[0,∞)(x). Then {ψjk:g∈L2(ℝ)}j∈ℤ,k∈ℕ0 is a Wilson frame for L2(ℝ).

(f) Let g(x)=w0(x)=w-1(x)=21/2/(1+2πix). Then {ψjk: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 {ψjk:w0,w-1∈L2(ℝ)}j∈ℤ,k∈ℕ0 is a Wilson frame for L2(ℝ).

(h) Let
(8)g(x)={1+x,if x∈[0,1),x2,if x∈[1,2),0,otherwise.

Then {ψjk: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 {ψjk: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),
(9)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=2∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw0(·)〉|211111111+|〈f,sin(2kπ·)Tjw-1(·)〉|2).

Proof.
Let f∈L2(ℝ). Then
(10)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2 =∑j:evenk∈ℕ0|∫f(x)εkcos(2kπx)w0(x-j2)¯dx|2 +∑j:oddk∈ℕ0|∫2f(x)sin(2(k+1)πx)w-1(x-j+12)¯dx|2.
This gives
(11)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=2∑j∈ℤ|∫f(x)w0(x-j)¯dx|2 +4∑j∈ℤk∈ℕ|∫f(x)cos(2kπx)1111111111×w0(x-j)¯dx∫f(x)sin(2kπx)|2 +4∑j∈ℤk∈ℕ|∫f(x)sin(2kπx)1111111111×w-1(x-j)¯dx∫f(x)sin(2kπx)|2.
Thus,
(12)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=2∑j∈ℤ|∫f(x)Tjw0(x)¯dx|2 +4∑j∈ℤk∈ℕ|〈f,cos(2kπ·)Tjw0(·)〉|2 +4∑j∈ℤk∈ℕ|〈f,sin(2kπ·)Tjw-1(·)〉|2=2∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw0(·)〉|2111111111+|〈f,sin(2kπ·)Tjw-1(·)〉|2).

Lemma 9.
Let {ψjk: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),
(13)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=∑j,k∈ℤ|〈f,EkTjw0〉|2+∑j,k∈ℤ|〈f,EkTjw-1〉|2 +∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw0(·)〉|2111111111+|〈f,sin(2kπ·)Tjw-1(·)〉|2) -∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw-1(·)〉|2111111111+|〈f,sin(2kπ·)Tjw0(·)〉|2).

Proof.
We have
(14)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=∑j:evenk∈ℕ0|〈f,ψjk〉|2+∑j:oddk∈ℕ0|〈f,ψjk〉|2=2∑j∈ℤ|〈f,Tjw0〉|2 +∑j∈ℤk∈ℕ|〈f,EkTjw0+E-kTjw0〉|2 +∑j∈ℤk∈ℕ|〈f,EkTjw-1-E-kTjw-1〉|2=2∑j∈ℤ|〈f,Tjw0〉|2+∑j,k∈ℤk≠0|〈f,EkTjw0〉|2 +∑j,k∈ℤk≠0|〈f,EkTjw-1〉|2 +2Re∑j,k∈ℤk≠0〈f,EkTjw0〉〈f,E-kTjw0〉¯ -2Re∑j,k∈ℤk≠0〈f,EkTjw-1〉〈f,E-kTjw-1〉¯.
Therefore, using
(15)Re〈f,EkTjg〉〈f,E-kTjg〉¯=(|〈f,cos(2kπ·)Tjg(·)〉|21111-|〈f,sin(2kπ·)Tjg(·)〉|2),
we finally get the result.

Remark 10.
Combining Lemmas 8 and 9, we get
(16)∑j,k∈ℤ|〈f,EkTjw0〉|2+∑j,k∈ℤ|〈f,EkTjw-1〉|2=∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw-1(·)〉|211111111+|〈f,sin(2kπ·)Tjw0(·)〉|2) +∑j,k∈ℤ(|〈f,cos(2kπ·)Tjw0(·)〉|21111111111+|〈f,sin(2kπ·)Tjw-1(·)〉|2).

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 {EkTjw0}k,j∈ℤ and {EkTjw-1}k,j∈ℤ are Bessel sequences.

The following result gives a necessary condition for a Wilson system {ψjk: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 {ψjk: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
(17)A2≤∑j∈ℤ(|w0(x-j)|2+|w-1(x-j)|2).
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
(18)W(x)=∑j∈ℤ(|w0(x-j)|2+|w-1(x-j)|2)<A2 on Δ.
We can assume that this Δ is contained in an interval of length 1. Let
(19)Δ0={x∈Δ:W(x)≤A2-1},Δk={x∈Δ:A2-1k<W(x)<A2-1k+1}.
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 Δk′. Choose f=χΔk′. Then ∥f∥=|χΔk′|, where |χΔk′| = measure of χΔk′.

Since for j∈ℤ, the functions fTjw0¯ and fTjw-1¯ have support in Δk′, {Ek}k∈ℤ constitute an orthonormal basis for L2(I), for every interval I of length 1 and Δk′ is contained in an interval of length 1, we have
(20)∑k∈ℤ|〈f,EkTjw0〉|2=∑k∈ℤ|〈fTjw0¯,Ek〉|2=∫ℝ|f(x)|2|w0(x-j)|2dx.
Also, since f=χΔk′, we have
(21)∑j,k∈ℤ|〈f,EkTjw0〉|2=∑j∈ℤ∫Δk′|w0(x-j)|2dx.
Similarly, we obtain
(22)∑j,k∈ℤ|〈f,EkTjw-1〉|2=∑j∈ℤ∫Δk′|w-1(x-j)|2dx.
Therefore,
(23)∑j,k∈ℤ(|〈f,EkTjw0〉|2+|〈f,EkTjw-1〉|2) =∑j∈ℤ∫Δk′(|w0(x-j)|2+|w-1(x-j)|2)dx =∫Δk′W(x)dx.
Further, since
(24)∑j∈ℤ,k∈ℕ0|〈f,ψjk〉|2 ≤2∑j,k∈ℤ(|〈f,EkTjw0〉|2+|〈f,EkTjw-1〉|2),
we get
(25)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2≤2∫Δk′W(x)dx≤2(A2-1k′+1)∫Δk′dx=(A-2k′+1)∥f∥2.
Hence,
(26)∑j,k∈ℤ|〈f,ψjk〉|2<A∥f∥2.
This is a contradiction.

Next, we give a sufficient condition for a Wilson system {ψjk:w0,w-1∈L2(ℝ)}j∈ℤ,k∈ℕ0 to be a Wilson Bessel sequence.

Theorem 13.
Let w0,w-1∈L2(ℝ),
(27)B1=supx∈[0,1]∑k∈ℤ|∑n∈ℤw0(x-n)w0(x-n-k)¯|<∞,B2=supx∈[0,1]∑k∈ℤ|∑n∈ℤw-1(x-n)w-1(x-n-k)¯|<∞.
Then, the Wilson system {ψjk: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 {EkTjw0}k,j∈ℤ and {EkTjw-1}k,j∈ℤ are Gabor Bessel sequences. Therefore, using Lemma 8 and Remark 10, the Wilson system {ψjk: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 {ψjk: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 {ψjk: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 {ψjk: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 {ψjk: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 {EkTjw0}k,j∈ℤ and {EkTjw-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 {ψjk: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 {ψjk:w0∈L2(ℝ)}j∈ℤ,k∈ℕ0 to be a Wilson frame.

Theorem 16.
Let w0∈L2(ℝ) and
(28)B=supx∈[0,1]∑k∈ℤ|∑n∈ℤw0(x-n)w0(x-n-k)¯|<∞.
Then the Wilson system {ψjk:w0∈L2(ℝ}j∈ℤ,k∈ℤ0 is a Wilson Bessel sequence. Further, if
(29)A=infx∈[0,1][∑k≠0|∑n∈ℤw0(x-n)w0(x-n-k)¯|∑n∈ℤ|w0(x-n)|2 -∑k≠0|∑n∈ℤw0(x-n)w0(x-n-k)¯|]>0,
then the Wilson system {ψjk: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 {EkTjw0}k,j∈ℤ is a Gabor frame for L2(ℝ). If we choose w0=w-1 in Lemma 9, then
(30)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=2∑j,k∈ℤ|〈f,EkTjw0〉|2 ∀f∈L2(ℝ).
Hence, the Wilson system {ψjk: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 {ψjk: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 {ψjk: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 {ψjk: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 {ψjk: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 {EkTjw0}k,j∈ℤ can never become a frame, and hence, the Wilson system {ψjk:w0∈L2(ℝ}j∈ℤ,k∈ℕ0 is not a Wilson frame.

Finally, we give a necessary and sufficient condition for a Wilson system {ψjk: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 {ψjk: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
(31)∑j∈ℤk∈ℕ0|〈f,ψjk〉|2=2∑j,k∈ℤ|〈f,EkTjw0〉|2 ∀f∈L2(ℝ).
Therefore, the Wilson system {ψjk: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 {EkTjw0}k,j∈ℤ is a frame for L2(ℝ) with bounds A and B. Since, the Gabor system {EkTjw0}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.