Multiresolution analysis on spectra of hermitian matrices

TL;DR

This paper develops a multiresolution analysis framework for Hermitian matrices invariant under unitary conjugation, introducing wavelet bases and methods to construct radial scaling functions, with potential extensions to Lie groups.

Contribution

It establishes a multiresolution analysis on Hermitian matrices respecting unitary invariance, characterizes orthonormal wavelet bases, and links classical scaling functions to the matrix setting.

Findings

Existence of orthonormal wavelet bases for $L^2( ext{Herm}(n))^{U(n)}$

Construction of radial scaling functions from classical ones on $\

Potential generalizations to compact Lie groups' Cartan decompositions.

Abstract

We establish a multiresolution analysis on the space $\text{Herm}(n)$ of $n\times n$ complex Hermitian matrices which is adapted to invariance under conjugation by the unitary group $U(n).$ The orbits under this action are parametrized by the possible ordered spectra of Hermitian matrices, which constitute a closed Weyl chamber of type $A_{n-1}$ in $\mathbb R^n.$ The space $L^2(\text{Herm}(n))^{U(n)}$ of radial, i.e. $U(n)$-invariant $L^2$-functions on $\text{Herm}(n)$ is naturally identified with a certain weighted $L^2$-space on this chamber. The scale spaces of our multiresolution analysis are obtained by usual dyadic dilations as well as generalized translations of a scaling function, where the generalized translation is a hypergroup translation which respects the radial geometry. We provide a concise criterion to characterize orthonormal wavelet bases and show that such bases always exist. They provide natural orthonormal bases of the space $L^2(\text{Herm}(n))^{U(n)}.$ Furthermore, we show how to obtain radial scaling functions from classical scaling functions on $\mathbb R^{n}$. Finally, generalizations related to the Cartan decompositions for general compact Lie groups are indicated.