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Are MSF wavelets minimally supported?

Marcin Bownik, Ziemowit Rzeszotnik, Darrin Speegle

TL;DR

The paper resolves a non-measurable variant of Larson's question by proving that the Fourier support $E$ of any dyadic wavelet necessarily contains a (possibly non-measurable) wavelet set. It connects wavelet sets, MSF wavelets, and operator-theoretic constructions with tiling concepts, leveraging Isbell's diagonal theorem for doubly stochastic matrices to extract a subset within $E$ that tiles under both translations and dilations. The results highlight a geometric structure in $\operatorname{supp}\hat{\psi}$ and establish that a non-measurable wavelet-set inside the support always exists, while measurability of such a set remains open in general. The work also discusses Steinhaus-type tiling problems, MRA-related observations, and related measurable-subset results, framing avenues for stronger, dimension-function–preserving questions.

Abstract

Larson's problem asks ``Must the support of the Fourier transform of a wavelet contain a wavelet set?". We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson's problem and propose two new related problems.

Are MSF wavelets minimally supported?

TL;DR

The paper resolves a non-measurable variant of Larson's question by proving that the Fourier support of any dyadic wavelet necessarily contains a (possibly non-measurable) wavelet set. It connects wavelet sets, MSF wavelets, and operator-theoretic constructions with tiling concepts, leveraging Isbell's diagonal theorem for doubly stochastic matrices to extract a subset within that tiles under both translations and dilations. The results highlight a geometric structure in and establish that a non-measurable wavelet-set inside the support always exists, while measurability of such a set remains open in general. The work also discusses Steinhaus-type tiling problems, MRA-related observations, and related measurable-subset results, framing avenues for stronger, dimension-function–preserving questions.

Abstract

Larson's problem asks ``Must the support of the Fourier transform of a wavelet contain a wavelet set?". We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson's problem and propose two new related problems.

Paper Structure

This paper contains 3 sections, 11 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Non-measurable Larson's problem

Key Result

Theorem 2.1

Let $\psi$ be a function in $L^2(\mathbb{R})$ such that $|\hat{\psi}| = \mathbf 1_E$. The function $\psi$ is a wavelet if and only if $E$ is a wavelet set. In particular, if $W$ is a wavelet set and $m: W \to \mathbb{C}$ is a unimodular function, then $\psi$ given by $\hat{\psi}(\xi)=m(\xi) \mathbf

Theorems & Definitions (25)

  • Theorem 2.1
  • Example 2.2
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Remark 2.6
  • Corollary 2.7
  • proof
  • Example 2.8
  • Proposition 2.10
  • ...and 15 more