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arXiv · 2307.13481

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

Nicki Holighaus·Günther Koliander
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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.

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Nicki Holighaus, Günther Koliander. 2023-07-25. Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems. https://arxiv.org/abs/2307.13481

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