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

2022/11/18 by Boris Rubin, Rubin, Boris · 2 citations
Mathematics · Medicine · Physics and Astronomy · #44A12 #Advanced Differential Geometry Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications

paper · pdf · doi:10.48550/arxiv.2211.10348

openalex publication_date 2022/11/18 · openalex created_date 2022/11/28 · openalex updated_date 2026/07/28

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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