On the existence of Riesz bases of exponentials

CARLOS CABRELLI; HARE, K.; URSULA M. MOLTER

Congreso; pecial Session of the AMS, Special Session on Analysis, Combinatorics, and Geometry of Fractals; 2021

American MAthematical Society

We address the following question: Given $\Omega$, a subset of $\mathbb{R}^d$ of finite measure, does there exist a discrete set $\mathcal{B}$ in $\mathbb{R}^d$ such that the exponential functions ${E}(\mathcal{B}) = \{e^{2\pi i\beta\cdot \omega}: \beta \in \mathcal{B}\}$ form a Riesz basis of $L^2(\Omega)$?Using the Bohr compactification of the group of integers,\ we are able to find necessary and sufficient condition to ensure that a {\em multi-tile} $\Omega \subset \mathbb{R}^{d}$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $L^{2}(\Omega )$. The main novelty is the necessity of the condition, since before sufficient conditions had been derived. Recall that a set $\Omega \subset \mathbb{R}^{d}$ is a $k$-multi-tile for $\mathbb{Z}^d$ if $\sum_{\lambda \in \mathbb{Z}^d} \chi_{\Omega}(\omega - \lambda) = k \ {\rm a.e.}\ \omega \in \mathbb{R}^d$.\