Multiresolution analysis on spectra of hermitian matrices

TL;DR

This work develops a multiresolution framework for $L^2(Herm(n))^{U(n)}$, exploiting the $U(n)$-invariance to reduce the analysis to the Weyl chamber and a hypergroup structure on spectra. By formulating radial translations, dyadic dilations, and a radial scaling function, the authors derive a two-scale relation and a practical criterion for constructing orthonormal wavelet bases, proving the existence of exactly $2^n-1$ radial wavelets. They connect radial scaling functions to classical, permutation-invariant scaling functions on $\mathbb{R}^n$ and provide explicit radial Shannon-type wavelets as a concrete example. The methodology generalizes to Cartan decompositions of compact Lie groups, offering a broad, high-rank extension of previous non-Euclidean radial wavelet constructions and reducing computational complexity from $2^{n^2}$ to $2^n$ factors in the radial setting.

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.

Theorems & Definitions (31)

• Lemma 2.1
• proof
• Lemma 2.2: graczykLoeb, Prop. 2
• Proposition 2.3
• proof
• Remark 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Definition 3.1