Radon transform on symmetric matrix domains

TL;DR

This paper develops explicit inversion formulas for the Radon transform on symmetric matrix domains over real, complex, and quaternionic fields, using invariant differential operators to facilitate integral geometry analysis.

Contribution

It introduces new inversion formulas for the Radon transform on symmetric matrix domains, employing explicit invariant differential operators for the first time in this context.

Findings

Derived explicit inversion formulas for the Radon transform.

Constructed invariant differential operators for the inversion process.

Applied formulas to symmetric matrix domains over various fields.

Abstract

Let $\bbK=\mathbb R, \mathbb C, \mathbb H$ be the field of real, complex or quaternionic numbers and $M_{p, q}(\bbK)$ the vector space of all $p\times q$-matrices. Let $X$ be the matrix unit ball in $M_{n-r, r}(\bbK)$ consisting of contractive matrices. As a symmetric space, $X=G/K=O(n-r, r)/O(n-r)\times O(r)$, $U(n-r, r)/U(n-r)\times U(r)$ and respectively $Sp(n-r, r)/Sp(n-r)\times Sp(r)$. The matrix unit ball $y_0$ in $M_{r^\prime-r, r}$ with $r^\prime \le n-1$ is a totally geodesic submanifold of $X$ and let $Y$ be the set of all $G$-translations of the submanifold $y_0$. The set $Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $\mathcal Rf(y)$ for functions $f\in C_0^\infty(X)$ defined by integration of $f$ over the subset $y$, and the dual transform $\mathcal R^t F(x), x\in X$ for functions $F(y)$ on $Y$. We find inversion formulas by constructing explicit certain invariant differential operators.