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

Published: 21 May 2023, Last Modified: 29 Aug 2023SampTA 2023 PaperReaders: Everyone
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 stable set of sampling for the wavelet system generated by $\psi$, for all $0<\beta\leq \beta(A,\psi)$. The result and choice of 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.
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