Wavelet Riesz bases associated to nonisotropic dilations

Abstract

A bounded, Riemann integrable and measurable set $K\subset {\mathbb{R}}^d$, which fulfills \[\sum\limits_{γ\inΓ}\mathbb{1}_K(x-γ)=k\text{ almost everywhere, $x\in {\mathbb{R}}^d$}\] 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.

14 pages, 3 figure