Frame spectral pairs and exponential bases

Abstract

Given a domain $\Omega \subset {\mathbb{R}}^d$ with positive and finite Lebesgue measure and a discrete set $\Lambda \subset {\mathbb{R}}^d$, we say that $(\Omega ,\Lambda )$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal{E}(\Lambda ):=\{e^{2\pi i\lambda \cdot x}:\lambda \in \Lambda \}$ is a frame for $L^2(\Omega )$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting ${\mathbb{Z}}_N^d$, $d,N\ge 1$, a frame spectral pair can be similarly defined. %(Here, ${\mathbb{Z}}_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs in ${\mathbb{R}}^d$ by "adding" frame spectral pairs in ${\mathbb{R}}^d$ and ${\mathbb{Z}}_N^d$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.