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

The goal of this paper is to describe the $α$-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 $α$ the $α$-cosine transform is a composition of the $(α+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $α$ except one we interpret the $α$-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 $α$, which is $-(min\{i,n-i\}+1)$, is still an open problem.