Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

Dae Gwan Lee; Goetz E. Pfander; David Walnut

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arXiv · 2401.08042

Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

Dae Gwan Lee·Goetz E. Pfander·David Walnut

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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.

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BibTeX RIS

Dae Gwan Lee, Goetz E. Pfander, David Walnut. 2024-01-16. Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice. https://arxiv.org/abs/2401.08042

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