Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals

Christina Frederick, Karamatou Yacoubou Djima

Research output: Contribution to journalArticlepeer-review

Abstract

A well-studied problem in sampling theory is to find an expansion of a function in terms of a Riesz basis of exponentials for L2(Ω), where Ω is a bounded, measurable set. For such a basis, we are guaranteed the existence of a unique biorthogonal dual basis that can be used to calculate the expansion coefficients. Much attention has been paid to the existence of Riesz bases of exponentials for various domains; however, the sampling and reconstruction problems in these cases are less understood. Recently, explicit formulas for the corresponding dual Riesz bases were introduced in Frederick and Okoudjou in [Appl Comput Harmon Anal 51:104–117, 2021; Frederick and Mayeli in J Fourier Anal Appl 27(5):1–21, 2021] for a class of multi-tiling domains. In this paper, we further this work by presenting explicit examples of a finite co-measurable union of intervals or multi-rectangles. In the higher-dimensional case, we also discuss how different sampling strategies lead to different Riesz bounds.

Original languageEnglish (US)
Pages (from-to)108-123
Number of pages16
JournalMatematica
Volume3
Issue number1
DOIs
StatePublished - Mar 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • Paley Wiener spaces
  • Riesz bases of exponentials
  • Vandermonde matrices

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