On an analytic description of the $\alpha$-cosine transform on real Grassmannians

TL;DR

This paper provides an explicit analytic description of the $\alpha$-cosine transform on real Grassmannians, relating it to Radon transforms and differential operators for most values of $\alpha$.

Contribution

It explicitly characterizes the $\alpha$-cosine transform as compositions of known transforms and differential operators, except for one unresolved case.

Findings

Most $\alpha$-cosine transforms are compositions of Radon transforms and differential operators.

Explicit formulas are provided for all but one special value of $\alpha$.

The case for the last remaining $\alpha$ value remains an open problem.

Abstract

The goal of this paper is to describe the $\alpha$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $\alpha$ the $\alpha$-cosine transform is a composition of the $(\alpha+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $\alpha$ except one we interpret the $\alpha$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $\alpha$, which is $-(min\{i,n-i\}+1)$, is still an open problem.