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arXiv · 2211.10348

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

Boris Rubin
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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.

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Boris Rubin. 2022-11-18. On the Injectivity of the Shifted Funk-Radon Transform and Related Harmonic Analysis. https://arxiv.org/abs/2211.10348

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