# On Wilson bases in L2(R^d)

## TL;DR

This paper extends the construction of orthonormal Wilson bases from one dimension to multiple dimensions, starting from tight Gabor frames with higher redundancies, thereby generalizing existing results in time-frequency analysis.

## Contribution

It introduces a method to construct multi-dimensional orthonormal Wilson bases from tight Gabor frames with redundancy 2^k, expanding the known existence results.

## Key findings

- Constructed multi-dimensional Wilson bases from tight Gabor frames.
- Generalized the existence of Wilson bases to higher dimensions.
- Extended known results in time-frequency analysis.

## Abstract

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. 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 show that one can construct multi-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$ where $k=1, 2, ..., d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.