On Wilson bases in $L^2(\mathbb{R}^d)$

Marcin Bownik, Mads Sielemann Jakobsen, Jakob Lemvig, Kasso A. Okoudjou

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Abstract

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. It is well known that, starting from a tight Gabor frame for $L^{2}(\mathbb{R})$ with redundancy 2, one can construct an orthonormal Wilson basis for $L^2(\mathbb{R})$ whose generator is well localized in the time-frequency plane. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. Specifically, we show that one can construct $d$-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$, where $k=1, 2, \hdots, d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.

Original languageEnglish
JournalS I A M Journal on Mathematical Analysis
Volume49
Issue number5
Pages (from-to)3999-4023
ISSN0036-1410
DOIs
Publication statusPublished - 2017

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