Wavelets for non-expanding dilations and the lattice counting estimate

Marcin Bownik; Jakob Lemvig

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arXiv · 1601.07114

Wavelets for non-expanding dilations and the lattice counting estimate

Marcin Bownik·Jakob Lemvig

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Abstract

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix $A \in \mathrm{GL}(n,\mathbb{R})$. This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations $A$ with $\left|\det{A}\right|\ne 1$ and for almost every lattice $Γ$ with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and and almost every choice of a dilation $A$ with respect to the Haar measure on $\mathrm{GL}(n,\mathbb{R})$.

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Marcin Bownik, Jakob Lemvig. 2016-01-26. Wavelets for non-expanding dilations and the lattice counting estimate. https://arxiv.org/abs/1601.07114

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