On the Injectivity of the Shifted Funk-Radon Transform and Related Harmonic Analysis

TL;DR

This paper establishes necessary and sufficient conditions for the injectivity of the shifted Funk-Radon transform on the sphere, generalizing spherical means and developing new harmonic analysis tools involving Jacobi polynomials and Grassmannian harmonics.

Contribution

It introduces a comprehensive harmonic analysis framework for the shifted Funk-Radon transform, including new concepts of induced Stiefel harmonics and Funk-Hecke theorems, extending previous results.

Findings

Derived injectivity conditions using zeros of Jacobi polynomials

Developed new harmonic analysis tools for Grassmannian harmonics

Generalized spherical means on the sphere

Abstract

Necessary and sufficient conditions are obtained for injectivity of the shifted Funk-Radon transform associated with $k$-dimensional totally geodesic submanifolds of the unit sphere $S^n$ in $\mathbb{R}^{n+1}$. This result generalizes the well known statement for the spherical means on $S^n$ and is formulated in terms of zeros of Jacobi polynomials. The relevant harmonic analysis is developed, including a new concept of induced Stiefel (or Grassmannian) harmonics, the Funk-Hecke type theorems, addition formula, and multipliers. Some perspectives and conjectures are discussed.