Range characterization of Radon transforms on \(S^ n\) and \(P^ n\mathbb{R}\)
type: Item
instance of: scholarly article
title: Range characterization of Radon transforms on \(S^ n\) and \(P^ n\mathbb{R}\)
zbMATH Open document ID: 0806.53068
author: Tomoyuki Kakehi
published in: Journal of Mathematics of Kyoto University
publication date: 1995-02-19T00:00:00Z
review text: Let be \(R = R_ \ell\) the Radon transform on the \(n\)-dimensional sphere \(S^ n\), \((1 \leq \ell \leq n - 2\)), \(f\) an integrating function on \(S^ n\) and \(\xi \subset S^ n\) an oriented \(\ell\)-dimensional totally geodesic sphere. One defines \(R\) as follows \[ Rf(\xi) = {1\over \text{Vol}(S^ \ell)} \int_{x \in \xi} f(x) dv_ \xi(x), \] where \(dv_ \xi(x)\) is the canonical measure on \(\xi\). In this paper, the author proves the following theorem: There exists a fourth order invariant differential operator \(P\) on \(\widetilde{Gr}_{\ell + 1,n + 1}\) such that the range \(\text{Im }R\) of \(R\) is identical with the kernel \(\text{Ker }P\).
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 558894
zbMATH Keywords: Radon transform
reviewed by: Marius Stoka
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519262
OpenAlex ID: W1902538073
DOI: 10.1215/KJM/1250519262
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description: scientific article; zbMATH DE number 558894