Bimodal Wilson systems in
Introduction
Given that forms an orthonormal basis (ONB) for , it is easy to establish that is an ONB for , where is the characteristic function of . is the simplest example of Gabor systems, first introduced in 1946 by D. Gabor [12]. More generally, given and , the set is the Gabor system with generator (function) ϕ and (time-frequency) parameters . is called a Gabor frame if there exist such that for every we have A Gabor frame with 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, 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 and . The Gabor system is a tight frame for with frame bound if and only if ϕ satisfies a.e. for each .
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 . The following statements hold. For all , we have if and only if with convergence in , for all . If holds for all f in a dense subset , then this equality holds for all . Suppose is a Parseval frame. If for all j, then is an orthonormal basis for .
The characterization of the generators ϕ and the time-frequency parameters such that is a frame is still largely unresolved [14]. Nonetheless, it is known that if is a Gabor frame then . But when the system in (1.1) is never complete. Furthermore, is an ONB for if and only if . 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 is an ONB, then, the window ϕ must be poorly localized in time or frequency that is where 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 is a linear combination of two Gabor functions localized at and respectively. More precisely, given a Gabor system , the associated (bimodal) Wilson system is where With these notations, the following result was proved in [9]:
Theorem 1.3 [9] Let be such that and . Then the Gabor system is a tight frame for if, and only if, the Wilson system is an orthonormal basis for . Furthermore, one can choose with compact support.
Theorem 1.3 has been generalized from the case of Gabor frames on the separable lattice to non separable lattices where A is any invertible matrix such that , see [19], [23]. The underlying theme in all these results is a one-to-one association of a tight Gabor frame of redundancy 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 and . 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 were recently constructed starting from tight Gabor frame of redundancy for each , [5, Theorem 3.1 & Theorem 4.5].
In this paper, we show that starting from a tight Gabor frame of redundancy , 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 . There exists with such that the Gabor system is a tight frame for with frame bound if and only if the Wilson system is a Parseval frame for .
To convert this Wilson system into an ONB, one is left to normalize its elements to have unit 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 tight Gabor frame, the associated Wilson system should be made of linear combinations of 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 and can never lead to an ONB.
Theorem 1.5 Let . There exists no function with either compactly supported, or ϕ and having exponential decay, such that the Wilson system is an ONB for .
We recall that the space of smooth functions on with compact support is denoted by , the Schwartz class is , the space of tempered distributions is . The (unitary) Fourier transform is defined by with inverse given by The torus is denoted by . If , we define it's Fourier coefficients by
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 to be an ONB for . 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 . Finally, in Section 4 we use the Zak transform to construct more examples of generator such that ϕ and have exponential decay.
Section snippets
Characterization for Wilson bases in
In this section we find necessary and sufficient conditions on ϕ that guarantee that the Wilson system forms a Parseval frame, Theorem 2.1. In addition, by normalizing each vector in we find additional conditions needed to make this Parseval (Wilson) frame an ONB.
Theorem 2.1 Let , and is defined by (1.4). The following statements are equivalent: is a Parseval frame for . , and for each , where
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 have exponential decay. To achieve this we extended a construction originally given in [9] to the case of Gabor frame of redundancy when . The key tool needed to deal with this case is the Zak transform. Using this we have the following results.
Theorem 4.1 Let be real functions such that and 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
References (24)
- et al.
Weyl-Heisenberg frames, translation invariant systems and the Walnut representation
J. Funct. Anal.
(2001) Remarks on the local Fourier bases
Un principe d'incertitude fort en théorie du signal ou en mécanique quantique
C. R. Acad. Sci., Sér. 2
(1981)Heisenberg proof of the Balian-Low theorem
Lett. Math. Phys.
(1988)- et al.
Differentiation and the Balian-Low theorem
J. Fourier Anal. Appl.
(1995) - et al.
On Wilson bases in
SIAM J. Math. Anal.
(2017) - et al.
Des ondelettes pour détecter les ondes gravitationnelle
Gaz. Math.
(2016) Characterizations of Gabor systems via the Fourier transform
Collect. Math.
(2000)- et al.
A simple Wilson orthonormal basis with exponential decay
SIAM J. Math. Anal.
(1991) - et al.
Electromagnetic reflection–transmission problems in a Wilson basis: fiber-optic mode-matching to homogeneous media
Opt. Quantum Electron.
(2018)