Higher-rank wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices

Let Mn,m be the space of real n×m matrices which can be identified with the Euclidean space Rnm. We introduce continuous wavelet transforms on Mn,m with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on Mn,m and coincide with classical ones in the rank-one case m=1. We prove an analog of Calderón's reproducing formula for L2-functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on Mn,m. We also introduce continuous ridgelet transforms associated to matrix planes in Mn,m. An inversion formula for these transforms follows from that for the Radon transform. The new approach makes it possible to reconstruct a function on Rnm from data on a set of planes of zero measure.