Multiresolution analysis on spectra of hermitian matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909539266199552 |
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| author | Langen, Lukas Rösler, Margit |
| author_facet | Langen, Lukas Rösler, Margit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiresolution analysis on spectra of hermitian matrices Langen, Lukas Rösler, Margit Classical Analysis and ODEs Functional Analysis 42C40, 43A90 (Primary) 33C52, 43A85, 43A62 (Secondary) 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. |
| title | Multiresolution analysis on spectra of hermitian matrices |
| topic | Classical Analysis and ODEs Functional Analysis 42C40, 43A90 (Primary) 33C52, 43A85, 43A62 (Secondary) |
| url | https://arxiv.org/abs/2410.10364 |