Frame spectral pairs and exponential bases

TL;DR

This paper introduces a method to construct new frame spectral pairs in Euclidean space by combining existing pairs and explores their connection to sampling theory, unifying known examples of exponential frames.

Contribution

It presents a novel construction technique for frame spectral pairs in  and finite groups, linking spectral properties with sampling theory and unifying previous examples.

Findings

New classes of frame spectral pairs are constructed in  and finite groups.

The construction unifies exponential frames for unions of equal-volume cubes.

The work highlights the connection between spectral properties and sampling theory.

Abstract

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 similarly defined. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs 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. We also remark on the link between the spectral property of a domain and sampling theory.