Frame Spectral Pairs and Exponential Bases

Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say that (Ω, Λ) is a frame spectral pair if the set of exponential functions E(Λ) : = { e^2πiλ·x: λ∈ Λ} is a frame for L²(Ω). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting ZNd, d, N≥ 1 , a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in R^d by “adding” a frame spectral pair in R^d to a frame spectral pair in ZNd. 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.