Multiscaled wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

Olafsson, G.; Ournycheva, E.; Rubin, B.

Multiscaled wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

Authors

Olafsson, G.

Ournycheva, E.

Rubin, B.

Description

Let $M$ be the space of real $n\times m$ matrices which can be identified with the Euclidean space $R^{nm}$. We introduce continuous wavelet transforms on $M$ with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on $M$ and coincide with classical ones in the rank-one case $m=1$. We prove an analog of Calderon's reproducing formula for $L^2$-functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on $M$. We also introduce continuous ridgelet transforms associated to matrix planes in $M$. An inversion formula for these transforms follows from that for the Radon transform.
29 pages