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 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.
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The author is grateful to the referees for their suggestions and to Professor Mark Agranovsky for helpful inspiring discussions and sharing his knowledge of the subject.
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To the memory of Professor Lawrence Zalcman
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Rubin, B. On the injectivity of the shifted Funk–Radon transform and related harmonic analysis. JAMA (2024). https://doi.org/10.1007/s11854-024-0348-x
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DOI: https://doi.org/10.1007/s11854-024-0348-x