Strictly Expansive Matrices

TL;DR

This paper investigates properties of strictly expansive matrices, showing that certain classes are strictly expansive and others are not, with implications for wavelet construction.

Contribution

It establishes conditions under which integer matrices are strictly expansive and constructs wavelets with specific Fourier properties.

Findings

Strongly connected diagonally dominant integer matrices are strictly expansive.

Integer matrices with determinant two are not strictly expansive for certain sets.

Existence of orthonormal A-wavelets with compactly supported smooth Fourier transforms.

Abstract

If $A$ is an integer valued, strictly expansive matrix, then there exists an orthonormal $A$-wavelet whose Fourier transform is compactly supported and smooth. We show that strongly connected diagonally dominant integer matrices are strictly expansive, and that integer matrices with determinant two are not strictly expansive with respect to particularly nice sets.