Asymptotic formulas for the dual Radon transform and applications
type: Item
instance of: scholarly article
title: Asymptotic formulas for the dual Radon transform and applications
zbMATH Open document ID: 0598.44001
DOI: 10.1007/BF01161760
author: D. C. Solmon
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183696
review text: Asymptotic formulas for the behavior at infinity of \({\mathfrak R}^ tg(x)\) are given where \({\mathfrak R}^ t\) is the dual Radon transform and \(g\) is a function in the Schwartz space on \(S^{n-1}\times E^ 1\) (or the Hilbert transform of such a function). The coefficients in the asymptotic expansion involve the moments of \(g\) and hence the polynomial consistency conditions of \textit{S. Helgason} [Acta Math. 113, 153--180 (1965; Zbl 0163.16602)] on the range of the Radon transform. It is shown that \(g={\mathfrak R}f\) for some smooth \(f\) that is \(O(| x|^{-n-m-1})\), \(m\geq -1\), as \(| x| \to \infty\) if and only if \(g\) satisfies the first \(m+1\) conditions of Helgason. Inversion formulas for \({\mathfrak R}^ t\) and a related spherical transform are also given.
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 3964664
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: behavior at infinity
zbMATH Keywords: dual Radon transform
zbMATH Keywords: Hilbert transform
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: polynomial consistency conditions
zbMATH Keywords: inversion formulas
zbMATH Keywords: spherical transform
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3964664