Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

TL;DR

This paper explores conditions under which parallelepipeds in b^d can support Riesz bases of exponential functions with frequencies in a prescribed lattice, extending previous orthogonal basis results to more general bases.

Contribution

It establishes sufficient conditions for parallelepipeds to admit Riesz bases of exponentials with lattice-constrained frequencies, extending prior orthogonal basis criteria.

Findings

Provided a sufficient condition extending orthogonal basis criteria to Riesz bases.

Introduced a spectral norm constraint condition for the generating matrix of the parallelepiped.

Enhanced understanding of frequency-limited bases in multidimensional Fourier analysis.

Abstract

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 $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.