Analytic and Group-Theoretic Aspects of the Cosine Transform

TL;DR

This paper surveys recent advances in the analytic and group-theoretic properties of the cosine transform, extending classical results to higher-rank cases on Stiefel and Grassmann manifolds, with applications to integral geometry and representation theory.

Contribution

It introduces new results on the analytic continuation, polar sets, and spectral analysis of higher-rank cosine transforms, linking them with Fourier analysis and group representations.

Findings

Extended cosine transform theory to higher-rank manifolds

Established inversion formulas and spectral properties

Connected cosine transforms with group representations of SL(n, R)

Abstract

This is a brief survey of recent results by the authors devoted to one of the most important operators of integral geometry. Basic facts about the analytic family of cosine transforms on the unit sphere and the corresponding Funk transform are extended to the "higher-rank" case for functions on Stiefel and Grassmann manifolds. The main topics are the analytic continuation and the structure of polar sets, the connection with the Fourier transform on the space of rectangular matrices, inversion formulas and spectral analysis, and the group-theoretic realization as an intertwining operator between generalized principal series representations of SL(n, R).