On functions with zero integral moments over balls of fixed radius
type: Item
instance of: scholarly article
title: On functions with zero integral moments over balls of fixed radius
zbMATH Open document ID: 1403.53063
DOI: 10.1134/S1064562418030213
published in: Doklady Mathematics
publication date: 2018-09-13T00:00:00Z
review text: For a locally integrable function \(f\) on \({\mathbb R}^n\), consider the integrals over all balls of a fixed radius \(r\): \[ \int_{|x|\leq r} f(x+u) \,dx, \quad u \in {\mathbb R}^n. \] It is known that if all these integrals are 0, this does not guarantee that \(f = 0\), unless further conditions are satisfied. The authors consider the case where \(f\) has support in a ball \(B_R\) of radius \(R > r\), and the integrals of the function and its first moments over all balls of radius \(r\) contained in \(B_R\) vanish: \[ \int_{|x|\leq r} f(x+u) \,dx = \int_{|x|\leq r} x_j f(x+u) \,dx = 0, \quad j=1,\ldots,n. \] The question is whether this is sufficient to imply \(f=0\) in \(B_R\). The authors state that for \(R > 2r\), this is true. For \(R \leq 2r\), it holds under extra conditions. If \(f \in C^\infty\), it holds for \(R = 2r\) without the extra conditions. For the case \(R > 2r\), they also provide a very complicated inversion formula for recovering \(f\) from its integrals, based on series expansions.
reviewed by: Fritz Keinert
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 6935999
zbMATH Keywords: zero integrals
zbMATH Keywords: integrals over balls
author: Valeriĭ V. Volchkov
author: Vit. V. Volchkov
MaRDI profile type: Publication
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full work available at URL: s1064562418030213
OpenAlex ID: W2883328441
description: scientific article; zbMATH DE number 6935999