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A non-vanishing property for tensor products of wavelets

Quentin Rible, Stéphane Seuret

Abstract

We prove that, given a wavelet $ψ$, it is possible to choose some multi-integers $(p_j=(p_{j,1},...,p_{j,d}))_{j \in \mathbb{Z}} \in \mathbb{Z}^d$ such that, for every $x=(x_1,...,x_d) \in \mathbb{R}^d$, for infinitely many integers $j$, the tensorized wavelet $\prod_{i=1}^d ψ(2^j x_i-p_{j,i})$ does not vanish at $x$. This non-vanishing property is essential for analyzing some generic regularity properties in certain Sobolev and Besov spaces. The proof relies on an assumption regarding the zeros of $ψ$, which we numerically verify for the first Daubechies wavelets.

A non-vanishing property for tensor products of wavelets

Abstract

We prove that, given a wavelet , it is possible to choose some multi-integers such that, for every , for infinitely many integers , the tensorized wavelet does not vanish at . This non-vanishing property is essential for analyzing some generic regularity properties in certain Sobolev and Besov spaces. The proof relies on an assumption regarding the zeros of , which we numerically verify for the first Daubechies wavelets.
Paper Structure (3 sections, 5 theorems, 33 equations, 2 figures)

This paper contains 3 sections, 5 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theo:lower_bound_periodised_wavelet_function']}
  3. Numerical proof of (R)

Key Result

Theorem 1.2

Let $\psi:\mathbb{R}\to\mathbb{R}$ be a function satisfying property $(R)$, and let be the periodized version of $\psi$. There is $\alpha>0$ with the following property: for every $d \in \mathbb{N}^*$, there exists an integer $N(d)$ and a sequence of multi-integer $(\overline{p_j}=(p_{j,1},\ldots,p_{j,d}))_{j\in\mathbb{N}}$ such that for every $x\in\mathbb{R}^{d}$ and $J\in \mathbb

Figures (2)

  • Figure 3.1: Daubechies wavelets 'db3' and 'db7' verify $(R)$.
  • Figure 3.2: Function $S$ associated with 'db45': it is strictly positive.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 3 more