Riesz bases of exponentials and the Bohr topology

CABRELLI, CARLOS A.; HARE, KATHRYN; MOLTER, URSULA M.

College Park, Maryland

Seminario; Faraway Fourier Talks; 2020

Norbert Wiener Center

In this talk we address the question of what domains $\Omega$\ of $\mathbb{R}^d$ with finite measure, admit a Riesz basis\ of exponentials, that is, the existence of a discrete set $\mathcal{B}$ in $\mathbb{R}^d$ such that the exponentials ${E}(\mathcal{B}) = \{e^{2\pi i\beta\cdot \omega}: \beta \in \mathcal{B}\}$ form a Riesz basis of $L^2(\Omega).$Using the Bohr compactification of the integers,\ we show a necessary and sufficient condition to ensure that a {\em multi-tile} $\Omega \subset \mathbb{R}^{d}$ of positive measure (but notnecessarily bounded) admits a structured Riesz basis of exponentials for $L^{2}(\Omega )$. Here a set $\Omega \subset \mathbb{R}^{d}$ is a $k$-multi-tile for $\mathbb{Z}^d$ if $\sum_{\lambda \in \mathbb{Z}^d} \chi_{\Omega}(\omega - \lambda) = k \ {\rm a.e.}\ \omega \in \mathbb{R}^d$.\