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

TL;DR

This paper explores the analytic continuation of matrix cosine and sine transforms on Stiefel and Grassmann manifolds, revealing their connections to the Funk-Radon transform and identity operator, with new pointwise results and open problems.

Contribution

It extends previous distributional results to pointwise analysis and establishes new connections between transforms and integral geometry on manifolds.

Findings

Cosine transform at a=0 is proportional to Funk-Radon transform.

Sine transform at a=0 acts as the identity operator.

New problems related to the transforms are formulated.

Abstract

We investigate analytic continuation of the matrix cosine and sine transforms introduced in Part I and depending on a complex parameter $\a$. It is shown that the cosine transform corresponding to $\a=0$ is a constant multiple of the Funk-Radon transform in integral geometry for a pair of Stiefel (or Grassmann) manifolds. The same case for the sine transform gives the identity operator. These results and the relevant composition formula for the cosine transforms were established in Part I in the sense of distributions. Now we have them pointwise. Some new problems are formulated.