Higher-rank wavelet transforms, ridgelet transforms, and Radon transforms on the space of matrices
Let \(\mathfrak M_{n,m}\) be the space of real \(n\times m\) matrices which can be identified with the Euclidean space \(\mathbb R^{nm}\). We introduce continuous wavelet transforms on \(\mathfrak M_{n,m}\) with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on \(\mathfrak M_{n,m}\) and coincide with classical ones in the rank-one case \(m=1\). We prove an analog of \textit{A. P. Calderón's} [Stud. Math. 24, 113--190 (1964; Zbl 0204.13703)], \textit{M. Frazier, B. Jawerth} and \textit{G. Weiss} [Regional Conference Series in Mathematics 79 (1991; Zbl 0757.42006)], \textit{B. Rubin} [J. Fourier Anal. Appl. 4, No.~2, 175--197 (1998; Zbl 0933.42023), Frat. Calc. Appl. 3, No.~1, 103--106 (2000)] reproducing formula for \(L^2\)-functions and obtain explicit inversion formulas for the Riesz potentials and Radon transforms on \(\mathfrak M_{n,m}\). We also introduce continuous ridgelet transforms associated to matrix planes in \(\mathfrak M_{n,m}\). An inversion formula for these transforms follows from that for the Radon transform. The new approach makes it possible to reconstruct a function on \(\mathbb R^{nm}\) from data on a set of planes of zero measure.
scientific article; zbMATH DE number 5069085
scientific article