Functions with vanishing integrals over spheres centered on cones
type: Item
instance of: scholarly article
title: Functions with vanishing integrals over spheres centered on cones
zbMATH Open document ID: 1236.44004
DOI: 10.1134/S1064562411030094
author: Valeriĭ V. Volchkov
author: Vit. V. Volchkov
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: Suppose that \({\mathbb R}^n\) is Euclidean space of dimension \(\geq 2\) with Euclidean norm \(|\cdot |\), \(d\omega_{n-1}\) is the surface measure on the sphere \({\mathbb S}^{n-1}=\{x\in{\mathbb R}^n : |x|=1\}\) induced by the Lebesgue measure, and \(\omega_{n-1}\) is the area of \({\mathbb S}^{n-1}\). By \(L^{1,loc}({\mathcal U})\) we denote the class of locally integrable functions on the domain \({\mathcal U}\subset {\mathbb R}^n\). For any function \(f\in L^{1,loc}({\mathcal U})\), any \(x\in{\mathcal U}\), and almost all \(r\in (0, dist(x, \partial{\mathcal U}))\), the spherical mean operator \[ {\mathcal R} f(x,r)=\frac1{\omega_{n-1}}\int_{{\mathbb S}^{n-1}} f(x+r\eta)\, d\omega_{n-1}(\eta) \] is defined. This operator is also known as the Radon transform on spheres. Let \(X(\mathcal U)\) be a class of locally summable functions on \(\mathcal U\). The kernel of the transform \(\mathcal R\) on the class \(X(\mathcal U)\) with respect to a set \(E\subset{\mathcal U}\) is defined as the set of functions \(f\in X(\mathcal U)\) such that \({\mathcal R} f(x,r)=0\) for all \(x\in E\) and almost all \(r\in (0, dist(x, \partial{\mathcal U}))\). The set \(E\subset{\mathcal U}\) is called the injectivity set of the transform \(\mathcal R\) if the kernel of \(\mathcal R\) with respect to \(E\) contains only the zero function. The authors study some problems connected with the spherical mean operator \(\mathcal R\) for \(X(\mathcal U)= L^{1,loc}(K_R)\) and \(E=\partial K_r\) or \(E= \partial K_{r_1}\cup \partial K_{r_2}\), where \(K_\alpha\) is the circular cone with aperture \(2\alpha\). The main results of the paper are as follows: (i) The kernel of \(\mathcal R\) with respect to \(\partial K_r\) is described for the class \(L^{1,loc}(K_R)\), and a sharp uniqueness theorem for functions from this kernel is proved. (ii) Necessary and sufficient conditions under which \(\partial K_{r_1}\cup \partial K_{r_2}\) is an injectivity set of \(\mathcal R\) for \(L^{1,loc}(K_R)\…
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 5998394
zbMATH Keywords: Radon transform on spheres
zbMATH Keywords: spherical mean operator
zbMATH Keywords: injectivity set
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
cites work: Q3331943
cites work: Q4437947
cites work: Q3228825
cites work: Approximation by spherical waves in \(L^p\)-spaces
cites work: Injectivity sets for the Radon transform over circles and complete systems of radial functions
cites work: Conical Uniqueness Sets for the Spherical Radon Transform
cites work: Q5640160
cites work: Q5820721
full work available at URL: s1064562411030094
OpenAlex ID: W1967123991
description: scientific article; zbMATH DE number 5998394