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Coherent frames with zero Beurling density

Ingrid Beltita, Jordy Timo van Velthoven

Abstract

We show the existence of a coherent frame in the orbit of a connected, simply connected unimodular solvable Lie group of exponential growth for which the lower Beurling density of its index set is zero.

Coherent frames with zero Beurling density

Abstract

We show the existence of a coherent frame in the orbit of a connected, simply connected unimodular solvable Lie group of exponential growth for which the lower Beurling density of its index set is zero.

Paper Structure

This paper contains 10 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Coherent systems over point sets in amenable groups
  3. Square-integrable projective representations
  4. Beurling density
  5. Coherent frames
  6. Proof of the main result
  7. Restriction of regular representations
  8. Solvable Lie groups of exponential growth
  9. Proof of Theorem \ref{['thm:zerodensity_intro']}
  10. A class of examples

Key Result

Theorem 1.1

Let $G$ be a second countable unimodular amenable group. Let $(\pi, \mathcal{H}_{\pi})$ be an irreducible, square-integrable projective unitary representation of $G$ of formal degree $d_{\pi} > 0$. If there exist $\Gamma \subseteq G$ and $\eta \in \mathcal{B}_{\pi}$ such that $\pi (\Gamma) \eta$ is $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1: fuehr2022groups
  • proof : Proof of \ref{['thm:multiplicity']}
  • Example 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof