Funk, Cosine, and Sine Transforms on Stiefel and Grassmann manifolds

TL;DR

This paper extends the Funk, cosine, and sine transforms from the sphere to Stiefel and Grassmann manifolds, exploring their properties, relations, and inversion formulas in a more general geometric context.

Contribution

It introduces new theoretical results on these transforms on Stiefel and Grassmann manifolds, including composition formulas and explicit inversion methods.

Findings

Derived composition formulas for the transforms.

Established Fourier functional relations for homogeneous distributions.

Provided explicit inversion formulas for the transforms.

Abstract

The Funk, cosine, and sine transforms on the unit sphere are indispensable tools in integral geometry. They are also known to be interesting objects in harmonic analysis. The aim of the paper is to extend basic facts about these transforms to the more general context for Stiefel or Grassmann manifolds. The main topics are composition formulas, the Fourier functional relations for the corresponding homogeneous distributions, analytic continuation, and explicit inversion formulas.