Exponential bases for parallelepipeds with frequency sets lying in the integer lattice

Dae Gwan Lee, Goetz E. Pfander, David Walnut

Published 2024-01-16Version 1

The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$ is well understood since the 1950s. In this paper, we consider the restriction to integer frequencies. That is, we give sufficient, and some necessary criteria, for a parallelepiped in $\mathbb{R}^d$ to permit a Fourier basis (orthonormal or Riesz) for the respective space of square integrable functions whose frequency set is constrained to be a subset of $\mathbb{Z}^d$. A simple rescaling argument provides criteria on parallelepipeds to permit frequency sets contained in a prescribed set $a_1\mathbb{Z} \times a_2\mathbb{Z} \times \ldots \times a_d\mathbb{Z}$, a restriction relevant in many applications.