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

摘要

Let M_(n,m) be the space of real n x m matrices which can be identified with the Euclidean space R~(nm). 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 Calderon's reproducing formula for L~2-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~(nm) from data on a set of planes of zero measure.