Journal article
On Wilson bases in $L^2(\mathbb{R}^d)$
University of Oregon1
Norwegian University of Science and Technology2
Department of Applied Mathematics and Computer Science, Technical University of Denmark3
Mathematics, Department of Applied Mathematics and Computer Science, Technical University of Denmark4
University of Maryland, College Park5
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.
Language: | English |
---|---|
Publisher: | Society for Industrial and Applied Mathematics |
Year: | 2017 |
Pages: | 3999-4023 |
ISSN: | 10957154 and 00361410 |
Types: | Journal article |
DOI: | 10.1137/17M1122190 |
ORCIDs: | Lemvig, Jakob |