Zeta integrals and integral geometry in the space of rectangular matrices

TL;DR

This paper explores the relationship between zeta distributions and Radon transforms on rectangular matrix spaces, providing new proofs and insights into their Fourier transforms, Bernstein identities, and convolution properties.

Contribution

It introduces a novel proof of the Bernstein identity using the Cayley-Laplace operator and applies these findings to Radon transforms in matrix spaces.

Findings

New proof of Bernstein identity based on Cayley-Laplace operator

Explicit Fourier transform formula for zeta distributions

Analysis of convolutions and Riesz potentials in matrix spaces

Abstract

The paper is devoted to interrelation between the zeta distribution and the Radon transform on the space of $n \times m$ real matrices. We present a self-contained proof of the Fourier transform formula for this distribution. Our method differs from that of J. Faraut and A. Koranyi in the part related to justification of the corresponding Bernstein identity. We suggest a new proof of this identity based on explicit representation of the radial part of the Cayley-Laplace operator. We also study convolutions with normalized zeta distributions, and the corresponding Riesz potentials. The results are applied to investigation of Radon transforms on the space of rectangular matrices.