Ranges and inversion formulas for spherical mean operator and its dual
type: Item
instance of: scholarly article
title: Ranges and inversion formulas for spherical mean operator and its dual
zbMATH Open document ID: 0845.43005
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-09-08T00:00:00Z
review text: The spherical mean operator is defined for a function \(f\) on \(\mathbb{R}^{n + 1}\) by \[ R(f) (r,x) = \int_{S^n} f(r \eta, x + r \xi) d \sigma_n (\eta, \xi); \quad (r,x) \in \mathbb{R} \times \mathbb{R}^n, \] where \(S^n\) is the unit sphere in \(\mathbb{R}^{n + 1}\) and \(\sigma_n\) is the normalized surface measure on \(S^n\). Some results from harmonic analysis are established for the generalized Fourier transform associated with the operator \(R\). The operator \(R\) and its dual \(^tR\) are transmutation operators on some function spaces. The authors prove inversion formulas for \(R\), \(^tR\) and a Plancherel theorem for \(^tR\).
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 44A05
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 861856
zbMATH Keywords: integral transforms
zbMATH Keywords: Bessel functions
zbMATH Keywords: spherical mean operator
zbMATH Keywords: harmonic analysis
zbMATH Keywords: generalized Fourier transform
zbMATH Keywords: transmutation operators
zbMATH Keywords: inversion formulas
zbMATH Keywords: Plancherel theorem
author: M. Mounir Nessibi
author: L. T. Rachdi
author: Khalifa Trimèche
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1448
OpenAlex ID: W2077209680
DOI: 10.1006/JMAA.1995.1448
description: scientific article; zbMATH DE number 861856