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

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.