Frame spectral pairs and exponential bases

Abstract

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 L 2 (Ω). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting Z d N , d, N ≥ 1, a frame spectral pair can be defined in a similar way. We show how to construct and obtain a new frame spectral pair in R d by "adding" frame spectral pairs in R d and Z d N . 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.