Finding duality for Riesz bases of exponentials on multi-tiles

Christina Frederick, Kasso Okoudjou

Published 2019-10-21Version 1

It is known that if $\Omega \subset \mathbb{R}^{d}$ is bounded, measurable set that forms a $k-$tiling of $\mathbb{R}^d$ when translated by a lattice $L$, there exists a Riesz basis of exponentials for $L^{2}(\Omega)$ constructed using $k$ translates of the dual lattice $L^*$. In this paper we give an explicit construction of the corresponding bi-orthogonal dual Riesz basis. In addition, we extend the iterative sampling algorithm introduced in prior work to this multivariate setting.