Finding duality for Riesz bases of exponentials on multi-tiles

TL;DR

This paper provides an explicit construction of dual Riesz bases of exponentials for multi-tiling domains and extends an iterative reconstruction algorithm to this setting.

Contribution

It introduces a method to explicitly construct dual Riesz bases for multi-tiling domains and adapts an iterative reconstruction algorithm accordingly.

Findings

Explicit dual Riesz bases are constructed for multi-tiling domains.

The iterative reconstruction algorithm is successfully extended to this setting.

The approach enhances understanding of exponential bases 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 $\Lambda^*$. 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.