The $\lambda$-Cosine Transforms, Differential Operators, and Funk Transforms on Stiefel and Grassmann Manifolds

TL;DR

This paper introduces new invariant differential operators linked to $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds, providing novel inversion formulas and analyzing transforms over matrices of varying rank.

Contribution

It develops a new family of invariant differential operators that simplify $\lambda$-cosine transforms and offers new inversion formulas on Stiefel and Grassmann manifolds.

Findings

New invariant differential operators for $\lambda$-cosine transforms

Inversion formulas for these transforms on manifolds

Analysis of Funk-cosine transforms over lower-rank matrices

Abstract

We introduce a new family of invariant differential operators associated with $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the $\lambda$-cosine transforms and yield new inversion formulas. Intermediate Funk-cosine transforms corresponding to integration over matrices of lower rank are studied. The main tools are polar decomposition and Fourier analysis on matrix space.