Bimodal Wilson systems in $L^2(\mathbb R)$

TL;DR

This paper introduces bimodal Wilson systems derived from Gabor frames in $L^2(R)$, establishing conditions for tight frames and Parseval systems, and demonstrating limitations in constructing well-localized orthonormal bases.

Contribution

It defines bimodal Wilson systems from Gabor frames, characterizes when they form tight frames or Parseval systems, and shows the impossibility of certain orthonormal basis constructions.

Findings

Gabor system $ ext{G}(, al, et)$ is a tight frame iff Wilson system $ ext{W}(, al, et)$ is Parseval.

Constructed examples of smooth, rapidly decaying generators $$.

Proved the non-existence of well-localized orthonormal bases via Wilson frames when $3 mid et^{-1}$.

Abstract

Given a window $\phi \in L^2(\mathbb R),$ and lattice parameters $\alpha, \beta>0,$ we introduce a bimodal Wilson system $\mathcal{W}(\phi, \alpha, \beta)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(\phi, \alpha, \beta)$. For a class of window functions $\phi,$ we show that the Gabor system $\mathcal{G}(\phi, \alpha, \beta)$ is a tight frame of redundancy $\beta^{-1}$ if and only if the Wilson system $\mathcal{W}(\phi, \alpha, \beta)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $\phi$ are constructed. In addition, when $3\leq \beta^{-1}\in \mathbb N$, 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 $L^2(\mathbb R)$.