Zeta integrals and integral geometry in the space of rectangular matrices

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.