Frame Spectral Pairs and Exponential Bases

Christina Frederick, Azita Mayeli

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Given a domain Ω⊂ Rd with positive and finite Lebesgue measure and a discrete set Λ⊂ Rd, we say that (Ω, Λ) is a frame spectral pair if the set of exponential functions E(Λ) : = { e2πiλ·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 Rd by “adding” a frame spectral pair in Rd 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.

Original languageEnglish (US)
Article number75
JournalJournal of Fourier Analysis and Applications
Volume27
Issue number5
DOIs
StatePublished - Oct 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • General Mathematics
  • Applied Mathematics

Keywords

  • Exponential bases and sampling
  • Frames
  • Riesz bases

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