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引用次数: 0
摘要
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。
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.
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