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

TL;DR

This paper demonstrates that rotated time-frequency lattices can serve as stable sampling sets for continuous wavelet systems, with specific matrices ensuring stability for smooth, localized wavelets, inspired by low discrepancy sequence theory.

Contribution

It introduces a particular class of rotated lattices as stable sampling sets for wavelet systems, extending previous results with new matrix constructions and estimates.

Findings

Existence of a generating matrix A for stable sampling lattices.

Stable sampling sets are valid for all sufficiently smooth, localized wavelets.

The approach combines lattice point estimates with coorbit theory techniques.

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 $0<\beta\leq \beta(A,\psi)$. 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.