Finding duality for Riesz bases of exponentials on multi-tiles

Abstract

It is known that if $\Omega \subset {\mathbb{R}}^d$ belongs to a class of multi-tiling domains when translated by a lattice $\Lambda$, there exists a Riesz basis of exponentials for $L^2(\Omega )$ constructed using $k$ translates of the dual lattice ${\Lambda }^{\ast }$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.