Erdelyi-Kober integrals on the cone of positive definite matrices and Radon transforms on Grassmann manifolds

TL;DR

This paper introduces generalized Erdelyi-Kober fractional integrals on positive definite matrices and demonstrates their connection to Radon transforms on Grassmann manifolds, revealing a unified structure in these integral transforms.

Contribution

It extends Erdelyi-Kober integrals to matrix cones and links them to Radon transforms on Grassmannians through analytic continuation, unifying existing formulas.

Findings

Erdelyi-Kober integrals are generalized to matrix cones.

Radon transforms are represented as analytic continuations of Erdelyi-Kober integrals.

Different formulas for Radon transforms share a common structure.

Abstract

We introduce bi-parametric fractional integrals of the Erdelyi-Kober type that generalize known Garding-Gindikin constructions associated to the cone of positive definite matrices. It is proved that the Radon transform, which maps a zonal function on the Grassmann manifold $G_{n,m}$ of $m$-dimensional linear subspaces of $R^n$ into a function on the similar manifold $G_{n,k}$, $ 1\leq m<k \leq n-1$, is represented as analytic continuation of the corresponding Erdelyi-Kober integral. This result shows that different Grinberg-Rubin's formulas for such transforms [GR] have, in fact, a common structure.