Exponential Bases for Parallelepipeds with Frequencies Lying in a Prescribed Lattice

Kurzfassung/Abstract

The existence of a Fourier basis with frequencies in for the space of square integrable functions supported on a given parallelepiped in , has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in , 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.