A non-vanishing property for tensor products of wavelets

TL;DR

This paper establishes a non-vanishing property for tensor products of wavelets, showing that for any point in space, infinitely many scaled and shifted tensor wavelets do not vanish at that point, aiding in regularity analysis.

Contribution

It proves a non-vanishing property for tensor wavelets under certain zero assumptions, verified numerically for Daubechies wavelets, advancing wavelet regularity analysis.

Findings

Non-vanishing tensor wavelet property proven

Numerical verification for Daubechies wavelets

Implications for Sobolev and Besov space analysis

Abstract

We prove that, given a wavelet $\psi$, 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 $\prod_{i=1}^d \psi(2^j x_i-p_{j,i})$ 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 $\psi$, which we numerically verify for the first Daubechies wavelets.