Mean value integral equations and the Helmholtz equation (Q1923647)

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.