Bimodal Wilson systems in L2(R)

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Abstract

Given a window ϕL2(R), and lattice parameters α,β>0, we introduce a bimodal Wilson system W(ϕ,α,β) consisting of linear combinations of at most two elements from an associated Gabor G(ϕ,α,β). For a class of window functions ϕ, we show that the Gabor system G(ϕ,α,β) is a tight frame of redundancy β1 if and only if the Wilson system W(ϕ,α,β) is Parseval system for L2(R). Examples of smooth rapidly decaying generators ϕ are constructed. In addition, when 3β1N, we prove that it is impossible to renormalize the elements of the constructed Parseval Wilson frame so as to get a well-localized orthonormal basis for L2(R).

Introduction

Given that {e2πim:mZ} forms an orthonormal basis (ONB) for L2([0,1)), it is easy to establish thatG(χ,1,1)={χ[0,1)(j)e2πim:j,mZ}, is an ONB for L2(R), where χ[0,1) is the characteristic function of [0,1). G(χ,1,1) is the simplest example of Gabor systems, first introduced in 1946 by D. Gabor [12]. More generally, given α,β>0 and ϕL2(R), the setG(ϕ,α,β)={ϕj,m():=ϕ(βj)e2πiαm:j,mZ} is the Gabor system with generator (function) ϕ and (time-frequency) parameters α,β. G(ϕ,α,β) is called a Gabor frame if there exist 0<AB such that for every fL2(R) we haveAf2j,mZ|f,ϕj,m|2Bf2. A Gabor frame with A=B is called a tight Gabor frame. In this case the frame bound A will be referred to as the redundancy of the tight Gabor frame. If in addition, A=B=1 we call the system a Parseval (Gabor) frame. We recall the following well-known result that will be used in the sequel, see [6, Theorem 8.1], and [8, Theorem 3.1].

Proposition 1.1

Let ϕL2(R) and α,β>0. The Gabor system G(ϕ,α,β) is a tight frame for L2(R) with frame bound β1 if and only if ϕ satisfies mZϕˆ(ξαm)ϕˆ(ξ+β1kαm)=δk,0 a.e. for each kZ.

In addition, the following result about Parseval frames and ONBs will be used repeatedly, we refer to [16, Section 7.1] for details.

Proposition 1.2

Let {ej}j=1L2(R). The following statements hold.

  • (1)

    For all fL2(R), we havef2=j=1|f,ej|2 if and only iff=j=1f,ejej, with convergence in L2(R), for all fL2(R).

  • (2)

    Iff2=j=1|f,ej|2 holds for all f in a dense subset DL2(R), then this equality holds for all fL2(R).

  • (3)

    Suppose {ej:j=1,2,...} is a Parseval frame. If ejL2=1 for all j, then {ej:j=1,2,...} is an orthonormal basis for L2(R).

The characterization of the generators ϕ and the time-frequency parameters α,β such that G(ϕ,α,β) is a frame is still largely unresolved [14]. Nonetheless, it is known that if G(ϕ,α,β) is a Gabor frame then 0<αβ1. But when αβ>1 the system in (1.1) is never complete. Furthermore, G(ϕ,α,β) is an ONB for L2(R) if and only if αβ=1. For more details about these density results we refer to [13, Section 7.5], [15], and the references therein. It is also known that all Gabor ONB behave essentially like our first example in the sense that if G(ϕ,α,1/α) is an ONB, then, the window ϕ must be poorly localized in time or frequency that isR|x|2|ϕ(x)|2dx=orR|ξ|2|ϕˆ(ξ)|2dξ= where ϕˆ is the Fourier transform of ϕ. This is the Balian-Low Theorem (BLT) that imposes strict limits on Gabor systems that form an ONB [2], [3], [4], [20].

Introduced numerically by K. G. Wilson [22], the so-called generalized Warnnier functions have good time-frequency localization properties and thus are not subjected to the localization limits dictated by the BLT. Latter, Daubechies, Jaffard, and Journé formalized this definition and introduced what is now known as Wilson systems [9]. Wilson ONBs have played major roles in some recent applications, including the detection of the gravitational waves [7], [17], [18], or their use in electromagnetic reflection-transmission problems in fiber optics [11], [10].

We now define the Wilson system for which each element ψj,m is a linear combination of two Gabor functions localized at (j,m) and (j,m) respectively. More precisely, given a Gabor system G(ϕ,α,β), the associated (bimodal) Wilson system W(ϕ,α,β) isW(ϕ,α,β)={ψj,m:jZ,mN0} whereψj,m(x)={2βϕ2j,0(x)=2βϕ(x2βj)ifjZ,m=0,β[e2πiβjαmϕj,m(x)+(1)j+me2πiβjαmϕj,m(x)]if(j,m)Z×N. With these notations, the following result was proved in [9]:

Theorem 1.3 [9]

Let ϕL2(R) be such that ϕˆ(ξ)=ϕˆ(ξ) and ϕ2=1. Then the Gabor system G(ϕ,1,1/2) is a tight frame for L2(R) if, and only if, the Wilson system W(ϕ,1,1/2) is an orthonormal basis for L2(R). Furthermore, one can choose ϕC(R) with compact support.

Theorem 1.3 has been generalized from the case of Gabor frames on the separable lattice Z×12Z to non separable lattices AZ2 where A is any invertible matrix such that |detA|=1/2, see [19], [23]. The underlying theme in all these results is a one-to-one association of a tight Gabor frame of redundancy (αβ)1=2 with a bimodal Wilson basis. However, it is still unknown whether similar associations can be made starting from a tight Gabor frame of other redundancy. For example, Gröchenig in [13, p. 168] posed the problem of the existence and construction of a Wilson ONB starting from a tight Gabor frame with α=1 and β=1/3. This problem is still unresolved. However, Wojdyłło proved that taking linear combinations of three elements of a redundancy 3 tight Gabor frame results in a (trimodal) Parseval Wilson frame [24]. But the method developed was not constructive and it is not clear how to use it to produce an example of a well-localized window function ϕ. In higher dimensions, Wilson ONBs are usually constructed by taking tensor products of 1 dimensional Wilson ONBs. In this context, (non-separable) Wilson ONBs for L2(Rd) were recently constructed starting from tight Gabor frame of redundancy 2k for each k=0,1,2,,d, [5, Theorem 3.1 & Theorem 4.5].

In this paper, we show that starting from a tight Gabor frame of redundancy 1/β, one can construct a bimodal Parseval Wilson frame. Furthermore, we can choose the generator to be a Schwartz function. For example, as a consequence of some of our results we shall prove the following.

Theorem 1.4

Let β(0,1/2). There exists ϕS(R) with ϕˆCc(R) such that the Gabor system G(ϕ,1,β) is a tight frame for L2(R) with frame bound β1 if and only if the Wilson system W(ϕ,1,β) is a Parseval frame for L2(R).

To convert this Wilson system into an ONB, one is left to normalize its elements to have unit L2 norm. However, we prove that this is impossible in general as the normalization conditions needed to get an ONB are incompatible with the definition of the Wilson system we use. In particular, our results suggest that for a redundancy β1N tight Gabor frame, the associated Wilson system should be made of linear combinations of β1 elements from the Gabor frame. It follows that the bimodal Wilson system given by (1.4) where the coefficients in the linear combinations are the unimodular numbers e2πiβjαm and (1)j+me2πiβjαm can never lead to an ONB.

Theorem 1.5

Let 3β1N. There exists no function ϕL2(R) with either ϕˆ compactly supported, or ϕ and ϕˆ having exponential decay, such that the Wilson system W(ϕ,1,β) is an ONB for L2(R).

We recall that the space of smooth functions on R with compact support is denoted by Cc(R), the Schwartz class is S(R), the space of tempered distributions is S(R). The (unitary) L2 Fourier transform is defined byFf(w)=fˆ(w)=Rf(t)e2πitwdt,wR, with inverse given byF1f(x)=f(x)=Rf(w)e2πixwdw,xR. The torus {zC:|z|=1} is denoted by T. If fL1(T), we define it's Fourier coefficients byfˆ(m)=Tf(x)e2πimxdx,(mZ).

The rest of the paper is organized as follows. Section 2 contains the technical results needed to prove our main results. In particular, we derive necessary and sufficient conditions on ϕ for the {ψj,m} to be an ONB for L2(R). In Section 3 we state and prove one of our main results Theorem 3.1. In particular, we give necessary and sufficient conditions to turn a tight Gabor frame into a Parseval Wilson system. We also indicate under which extra condition this Wilson system becomes an ONB, and provide examples of generators ϕS(R). Finally, in Section 4 we use the Zak transform to construct more examples of generator ϕS(R) such that ϕ and ϕˆ have exponential decay.

Section snippets

Characterization for Wilson bases in L2(R)

In this section we find necessary and sufficient conditions on ϕ that guarantee that the Wilson system W(ϕ,α,β) forms a Parseval frame, Theorem 2.1. In addition, by normalizing each vector in W(ϕ,α,β) we find additional conditions needed to make this Parseval (Wilson) frame an ONB.

Theorem 2.1

Let α,β>0, and {ψj,m}jZ,mN0 is defined by (1.4). The following statements are equivalent:

  • (a)

    W(ϕ,α,β)={ψj,m}jZ,mN0 is a Parseval frame for L2(R).

  • (b)

    Φk(ξ)=δk,0a.e., and Δk(ξ)=0a.e. for each kZ, where{Φk(ξ)=mZϕˆ(ξαm)ϕˆ(

Parseval Wilson frames

In this section we connect Gabor tight frames to the Wilson systems we defined. In particular, one of our main result is Theorem 3.1 from which Theorem 1.4 follows.

The Zak transform and Wilson systems

In this section we construct example of generators ϕ that satisfy the hypothesis of Theorem 3.1 and such that ϕ and ϕˆ have exponential decay. To achieve this we extended a construction originally given in [9] to the case of Gabor frame of redundancy NN when N3. The key tool needed to deal with this case is the Zak transform. Using this we have the following results.

Theorem 4.1

Let ϕˆ be real functions such that |ϕˆ(ξ)|(1+|ξ|)1ϵ and β=1/(2n) where n is any odd natural number. Then the following are

Acknowledgments

D. G. B. is grateful to Professor Kasso Okoudjou for hosting and arranging research facilities at the University of Maryland. D. G. B. is thankful to SERB Indo-US Postdoctoral Fellowship (2017/142-Divyang G Bhimani) for the financial support. D.G.B. would like to express many thanks to Professor Pascal Aucher for sending his paper [1]. D.G.B. is also thankful to DST-INSPIRE and TIFR CAM for the academic leave. K. A. O. was partially supported by a grant from the Simons Foundation # 319197, the

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