Exponential Bases for Parallelepipeds with Frequencies Lying in a Prescribed Lattice

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.