Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems

Abstract

We provide an example for the generating matrix $A$ of a two-dimensional lattice $\Gamma =A{\mathbb{Z}}^2$, such that the following holds: For any sufficiently smooth and localized mother wavelet $\psi$, there is a constant $\beta (A,\psi )>0$, such that $\beta \Gamma \cap (\mathbb{R}\times {\mathbb{R}}^{+})$ is a set of stable sampling for the wavelet system generated by $\psi$, for all $0<\beta \le \beta (A,\psi )$. The result and choice of the generating matrix are loosely inspired by the studies of low discrepancy sequences and uniform distribution modulo $1$. In particular, we estimate the number of lattice points contained in any axis parallel rectangle of fixed area. This estimate is combined with a recent sampling result for continuous wavelet systems, obtained via the oscillation method of general coorbit theory.

arXiv comment5 pages, presented at SampTA 2023 (14th International conference on Sampling Theory and Applications), corrected a typographical error over the published version