Wavelet Riesz bases associated to nonisotropic dilations

TL;DR

This paper constructs wavelet Riesz bases, including the first shearlet Riesz basis, for $L^2( eals^2)$ using nonisotropic dilations and $k$-tiling sets, advancing wavelet theory and applications.

Contribution

It introduces a method to build generalized Riesz wavelet bases from $k$-tiling sets, including the first shearlet Riesz basis, using nonisotropic dilations.

Findings

Established a link between $k$-tiling sets and Riesz bases of exponentials.

Constructed the first shearlet Riesz basis.

Demonstrated the applicability of nonisotropic dilations in wavelet basis construction.

Abstract

A bounded, Riemann integrable and measurable set $K\subset \mathbb{R}^d$, which fulfills \[\sum\limits_{\gamma\in\Gamma}\mathbb{1}_K(x-\gamma)=k\text{ almost everywhere, $x\in\mathbb{R}^d$}\] for a lattice $\Gamma\subset\mathbb{R}^d$ is called $k$-tiling. If $K\subset\mathbb{R}^d$ is $k$-tiling $L^2(K)$ will admit a Riesz basis of exponentials. We use this result to construct generalized Riesz wavelet bases of $L^2(\mathbb{R}^2)$, arising from the action of suitable subsets of the affine group. One example of our construction is the first known shearlet Riesz basis.