Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Calderón sum formula for wavelet systems

Ulrik Enstad, Jordy Timo van Velthoven

TL;DR

The paper resolves a long-standing question about Calderón-type completeness for wavelet systems with general dilations. By embedding discrete wavelet systems into a group-theoretic framework on $G=\mathbb{R}^d\rtimes\langle A\rangle$ and analyzing density via covolume of hulls, the authors derive precise bounds relating the continuous and discrete transforms. Under a mild admissibility condition $\psi\in\mathcal{B}_{\pi}$, they prove that a Parseval discrete frame implies the corresponding continuous semi-discrete wavelet system is a tight frame and that the Calderón sum holds: $\sum_{j\in\mathbb{Z}} |\widehat{\psi}((A^t)^j\xi)|^2 = |\det(P)|$ a.e. $\xi$, while also deducing $|\det(A)| \neq 1$ in the Parseval case. The results extend Calderón-type conditions to arbitrary dilations, connect frame theory with covolume on amenable groups, and partially resolve a conjecture by Bownik and Lemvig, offering a general approach applicable to nonunimodular settings and quasi-lattices.

Abstract

We show that the Calderón sum formula for orthonormal wavelet bases holds for arbitrary dilation and translation matrices under a mild condition on the wavelet function. This partially solves a conjecture by Bownik and Lemvig.

On the Calderón sum formula for wavelet systems

TL;DR

The paper resolves a long-standing question about Calderón-type completeness for wavelet systems with general dilations. By embedding discrete wavelet systems into a group-theoretic framework on and analyzing density via covolume of hulls, the authors derive precise bounds relating the continuous and discrete transforms. Under a mild admissibility condition , they prove that a Parseval discrete frame implies the corresponding continuous semi-discrete wavelet system is a tight frame and that the Calderón sum holds: a.e. , while also deducing in the Parseval case. The results extend Calderón-type conditions to arbitrary dilations, connect frame theory with covolume on amenable groups, and partially resolve a conjecture by Bownik and Lemvig, offering a general approach applicable to nonunimodular settings and quasi-lattices.

Abstract

We show that the Calderón sum formula for orthonormal wavelet bases holds for arbitrary dilation and translation matrices under a mild condition on the wavelet function. This partially solves a conjecture by Bownik and Lemvig.
Paper Structure (9 sections, 13 theorems, 45 equations)

This paper contains 9 sections, 13 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Point sets in locally compact groups
  3. Delone sets
  4. The hull of a point set
  5. Covolume of point sets
  6. Frame bounds, covolume and the norm of wavelet transform
  7. Wavelet transform
  8. Frames
  9. Calderón's condition for arbitrary dilations

Key Result

Theorem 1.2

Let $A, P \in \mathrm{GL}(d, \mathbb{R})$. If $\psi \in \mathcal{B}_{\pi}$ and $\{\pi(A^j P k, A^j) \psi \}_{ j \in \mathbb{Z}, k \in \mathbb{Z}^d }$ is a Parseval frame for $L^2 (\mathbb{R}^d)$, then $\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$ is a tight frame for $L^2 (\mathbb{

Theorems & Definitions (23)

  • Conjecture 1.1: bownik2017waveletsbownik2020openspeegle2003existence
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 13 more