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Exponential Riesz bases in non-Archimedean locally compact Abelian groups

Aihua Fan, Shilei Fan

TL;DR

This work extends the spectral theory of exponential Riesz bases to non-Archimedean l.c.a. groups, proving that any compact open set Ω has a Riesz basis of exponentials E(Λ) in L^2(Ω) and that there exist bounded open sets for which no exponential Riesz basis exists. The authors develop a two-pronged approach: first, a p-adic constructive method using finite-discrete reductions and a Λ = D + L_γ decomposition to obtain Riesz bases for compact open sets; second, a nonexistence framework based on translation numbers and a localization argument that yields explicit counterexamples in Q_p and in general non-Archimedean l.c.a. groups. The results highlight the delicate interplay between group structure, open subgroups, and spectral bases in non-Archimedean harmonic analysis, and they open paths for further exploration of multi-tiling and Riesz-basis questions in higher-dimensional p-adic contexts.

Abstract

This paper establishes two fundamental results on the existence of exponential Riesz basis in non-Archimedean locally compact Abelian groups: the existence of Riesz basis of exponentials for all finite unions of balls and the non-existence of such basis for some bounded sets.

Exponential Riesz bases in non-Archimedean locally compact Abelian groups

TL;DR

This work extends the spectral theory of exponential Riesz bases to non-Archimedean l.c.a. groups, proving that any compact open set Ω has a Riesz basis of exponentials E(Λ) in L^2(Ω) and that there exist bounded open sets for which no exponential Riesz basis exists. The authors develop a two-pronged approach: first, a p-adic constructive method using finite-discrete reductions and a Λ = D + L_γ decomposition to obtain Riesz bases for compact open sets; second, a nonexistence framework based on translation numbers and a localization argument that yields explicit counterexamples in Q_p and in general non-Archimedean l.c.a. groups. The results highlight the delicate interplay between group structure, open subgroups, and spectral bases in non-Archimedean harmonic analysis, and they open paths for further exploration of multi-tiling and Riesz-basis questions in higher-dimensional p-adic contexts.

Abstract

This paper establishes two fundamental results on the existence of exponential Riesz basis in non-Archimedean locally compact Abelian groups: the existence of Riesz basis of exponentials for all finite unions of balls and the non-existence of such basis for some bounded sets.

Paper Structure

This paper contains 12 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. preliminary
  3. Riesz sequences and frames in Hilbert spaces
  4. Non-Archimedean locally compact Abelian groups
  5. The field of $p$-adic numbers
  6. Proof of Theorem \ref{['thm-main1']}
  7. Riesz basis for finite subsets of a discrete Abelian group
  8. Existence of exponential Riesz bases for compact open set in $\mathbb{Q}_p$
  9. Existence of exponential Riesz bases for compact open set in non-Archimedean l.c.a. groups
  10. Proof of Proposition \ref{['thm:criterion']} and Theorem \ref{['thm-main2-general']}
  11. Nonexistence of exponential Riesz bases in $\mathbb{Q}_p$
  12. Nonexistence of exponential Riesz bases in general non-Archimedean l.c.a. groups

Key Result

Theorem 1.1

Let $G$ be a non-Archimedean l.c.a. group. For any compact open subset $\Omega \subset G$, there exist a set $\Lambda\subset \widehat{G}$ of characters which forms a Riesz basis of $L^2(\Omega)$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2: Affine Stability
  • proof
  • ...and 8 more