A non-vanishing property for tensor products of wavelets

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.