{"title":"On the injectivity of the shifted Funk–Radon transform and related harmonic analysis","authors":"Boris Rubin","doi":"10.1007/s11854-024-0348-x","DOIUrl":null,"url":null,"abstract":"<p>Necessary and sufficient conditions are obtained for injectivity of the shifted Funk–Radon transform associated with <i>k</i>-dimensional totally geodesic submanifolds of the unit sphere <i>S</i><sup><i>n</i></sup> in ℝ<sup><i>n</i>+1</sup>. This result generalizes the well known statement for the spherical means on <i>S</i><sup><i>n</i></sup> 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.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal d'Analyse Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11854-024-0348-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.
我们得到了与ℝn+1 中单位球 Sn 的 k 维完全大地子球面相关的移位 Funk-Radon 变换的注入性的必要条件和充分条件。这一结果概括了关于 Sn 上球面手段的众所周知的陈述，并用雅可比多项式的零点来表述。相关的谐波分析得到了发展，包括诱导 Stiefel（或格拉斯曼）谐波的新概念、Funk-Hecke 型定理、加法公式和乘数。还讨论了一些观点和猜想。
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