Mean value integral equations and the Helmholtz equation
type: Item
instance of: scholarly article
title: Mean value integral equations and the Helmholtz equation
zbMATH Open document ID: 0872.39013
DOI: 10.1007/BF03322178
author: Marc Chamberland
published in: Results in Mathematics
publication date: 1997-09-16T00:00:00Z
review text: The author determines, with aid of spectral synthesis, all continuous functions which assume at each point of the real line or plane the integral mean of its values on a symmetric interval or circular disk of fixed radius, respectively. In the one dimensional case the solutions are exactly the examples found by \textit{W. Walter} [Result. Math. 26, No. 3-4, 399-402 (1994; Zbl 0838.39006)] and by \textit{W. Foerg-Rob} [Report of Meeting 1994, Aequationes Math. 49, 191-192, 195-196, 198-200 (1995)]. While this paper is about mean value properties over ``volumes'', \textit{K. Iwasaki} [Discrete Comput. Geom. 17, No. 2, 163-189 (1997; reviewed below)] offers very general results on functions having mean value properties over surfaces.
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 47A25
zbMATH DE Number: 933254
zbMATH Keywords: mean value integral equations
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: spectral synthesis
zbMATH Keywords: continuous functions
zbMATH Keywords: mean value properties
reviewed by: János Aczél
MaRDI profile type: Publication
cites work: Q3331506
cites work: Spectral synthesis and the Pompeiu problem
cites work: Counterexamples to a problem of L. Schwartz
cites work: Q4332838
cites work: Théorie générale des fonctions moyenne-périodiques
cites work: A functional equation of J. A. Baker involving integral means
cites work: Q3955832
cites work: Mean values and differential equations
cites work: Offbeat Integral Geometry
full work available at URL: https://doi.org/10.1007/bf03322178
OpenAlex ID: W2083956348
description: scientific article; zbMATH DE number 933254