Riesz bases of exponentials for multi-tiling measures

TL;DR

This paper investigates conditions under which multi-tiling measures admit Riesz bases of exponentials, with applications to geometric shapes like the square boundary, revealing limitations of certain natural candidate bases.

Contribution

It provides necessary and sufficient conditions for multi-tiling measures to admit Riesz bases of exponentials and applies these results to geometric examples such as the square boundary.

Findings

Multi-tiling measures can admit Riesz bases under specific conditions.

The square boundary, after rotation, is a multi-tiling measure but does not admit a Riesz basis of a particular form.

Certain natural candidate bases for the square boundary are ruled out.

Abstract

Let $G$ be a closed subgroup of ${\mathbb R}^d$ and let $\nu$ be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group $G^{\perp}$. We form a multi-tiling measure $\mu = \mu_1+...+\mu_N$ where $\mu_i$ is translationally equivalent to $\nu$ and different $\mu_i$ and $\mu_j$ have essentially disjoint support. We obtain some necessary and sufficient conditions for $\mu$ to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of $G = {\mathbb Z}\times {\mathbb R}$ and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups ${\mathbb Z}\times \{0\}$. This rules out a natural candidate of potential Riesz basis for the square boundary.