An Elementary Eigenvalue Criterion for One-Parameter Dilation Groups to Admit a Continuous Wavelet

TL;DR

This paper introduces a straightforward eigenvalue-based criterion to determine if a one-parameter dilation group admits a continuous wavelet, providing a complete characterization in two-dimensional space.

Contribution

It offers a simple, eigenvalue-based condition for identifying dilation groups that support continuous wavelets, especially complete in 2D.

Findings

Eigenvalues of the symmetric part determine wavelet admitance.

Complete characterization of dilation groups in R^2.

Criterion simplifies wavelet analysis for matrix groups.

Abstract

In this article, we present a simple criterion for checking whether a one-parameter matrix group of dilations admits a continuous wavelet. This criterion involves only checking that the eigenvalues of the symmetric part of the matrix have the same sign. In $R^2$ this criterion gives a complete characterization of such matrix groups.