Wavelets for non-expanding dilations and the lattice counting estimate

TL;DR

This paper explores the connection between wavelet existence for non-expanding dilations and lattice point counting in geometry of numbers, establishing new results for a broad class of dilations and lattices.

Contribution

It demonstrates that lattice counting estimates hold for non-expanding dilations and almost every lattice, leading to the existence of minimally supported frequency wavelets in these cases.

Findings

Lattice counting estimate holds for all dilations with |det A| ≠ 1.

MSF wavelets exist for almost every lattice and dilation.

Results extend wavelet theory to non-expanding dilations.

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 $\Gamma$ 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 $\mathrm{GL}(n,\mathbb{R})$.