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 Sⁿ in ℝⁿ⁺¹. This result generalizes the well known statement for the spherical means on Sⁿ 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.