Frame spectral pairs and exponential bases

Christina Frederick, Azita Mayeli

Published 2020-10-12Version 1

Given a domain $\Omega\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $\Lambda\subset \Bbb R^d$, 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 $\Bbb Z_N^d$, $d, N\geq 1$, a frame spectral pair can be defined in a similar way. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain a new frame spectral pair in $\Bbb R^d$ by "adding" frame spectral pairs in $\Bbb R^{d}$ and $\Bbb Z_N^d$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. In this paper, we will also obtain a connection between frame spectral pairs and the Whittaker-Shannon interpolation formula when the frame is an orthogonal basis.