Stiefel and Grassmann manifolds

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.CONTENTS 1. Introduction.2. Preliminaries. 3. The higher-rank Funk transform.4. Cosine and sine transforms.Composition formulas.5. Cosine transforms via the Fourier analysis.6. Normalized cosine and sine transforms.7. The method of Riesz potentials.8. Appendix.