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Unions of exponential Riesz bases

Dae Gwan Lee

Abstract

We develop new methods for constructing exponential Riesz bases by taking unions of exponential Riesz bases. These methods are based on taking unions of frequency sets and domains respectively and therefore allow easy construction. Together with examples that illustrate our methods, we also provide several examples showing the delicate nature of exponential Riesz bases.

Unions of exponential Riesz bases

Abstract

We develop new methods for constructing exponential Riesz bases by taking unions of exponential Riesz bases. These methods are based on taking unions of frequency sets and domains respectively and therefore allow easy construction. Together with examples that illustrate our methods, we also provide several examples showing the delicate nature of exponential Riesz bases.
Paper Structure (11 sections, 10 theorems, 59 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Related work on Questions \ref{['Q1']} and \ref{['Q2']}
  3. Main Results
  4. Organization of the paper
  5. Preliminaries
  6. Comparison with Related Work
  7. Proof of Theorem \ref{['thm:two-sets-nested-translation-by-a']}
  8. Proof of Theorem \ref{['thm:RB-union-two-sets-generalization']}
  9. Proof of Theorem \ref{['thm:RB-union-two-sets-generalization-converse']}
  10. Proof of Theorem \ref{['thm:two-sets-nested-translation-by-a-dimension-d']}
  11. Proof of Proposition \ref{['prop:characterization-based-on-Fourier-submatrices']}

Key Result

Theorem 1

Let $S_1, S_2 \subset \mathbb R$ be disjoint bounded measurable sets with $S_1 {+} a \subset S_2$ for some $a \in \mathbb R \backslash \{ 0 \}$. If there exist sets $\Lambda_1 \subset \mathbb R \backslash \frac{1}{a}\mathbb{Z}$ and $\Lambda_2 \subset \frac{1}{a}\mathbb{Z}$ with $\mathrm{dist} (\Lamb

Figures (1)

  • Figure 1: An example of sets $S_1$ and $S_2$ together with $\Lambda_1$ and $\Lambda_2$

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Example
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Proposition 6: see e.g., Lemma 8 in Le21
  • Theorem 7: Kadec's $1/4$ theorem, see e.g., p. 36 in Yo01
  • Proposition 8
  • Remark
  • ...and 8 more