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

Let M_{n, m} be the space of real n × m matrices which can be identified with the Euclidean space R^{n m}. We introduce continuous wavelet transforms on M_{n, m} with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on M_{n, 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 M_{n, m}. We also introduce continuous ridgelet transforms associated to matrix planes in M_{n, 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 R^{n m} from data on a set of planes of zero measure.