Multiresolution analysis on spectra of Hermitian matrices

We establish a multiresolution analysis on the space Herm ( n ) of n × 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 R n . The space L 2 ( Herm ( n ) ) U ( n ) of radial, i.e . U ( n ) -invariant L 2 -functions on 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 ( Herm ( n ) ) U ( n ) . Furthermore, we show how to obtain radial scaling functions from classical scaling functions on R n . Finally, generalizations related to the Cartan decompositions for general compact Lie groups are indicated.