Functions satisfying a weighted average property
Anil Kumar Bose · Transactions of the American Mathematical Society · 1965
Introduction.It is well known that a harmonic function f(x) =f(xx,x2,---,x") defined in a given region (open connected set) R of the n-dimensional euclidean space £" can be characterized by the following mean-value properties, namely,where B(x0, r) and S(x0, r) denote any ball and its surface with x0 for its center and radius r which lies in R; dp and der stand for the usual Lebesgue measures of B and S; V" and €l" denote the total measures of B and S. We use the letters x and y to denote the n-dimensional vectors (xx,x2,---,x") and (yi,y2r--,yn)-Generalization of the above mean-value properties has been made in two directions, namely, (I) in replacing the figures B and S by other figures, and (II) in replacing the Lebesgue measure by other measures.A number of authors have dealt with generalization (I).In the case of two dimensions, it has been proved by J. L. Walsh [11], E. F. Beckenbach and M. O.Reade [10], A. Friedman [7] and others that if the figures B and S are replaced, respectively, by a regular polygon of N sides and its boundary, then the functions having the mean-value properties (0.1) and (0.2) are precisely the harmonic polynomials of degree ^ N having zero JVth derivatives in each of the directions of the radii of the polygon.Recently, in two very nice papers, L. Flatto [4] and A. Friedman and W. Liftman [6] have generalized the above mean-value properties (0.1) and (0.2) simultaneously in the two directions (I) and (II).They were interested in characterizing the class of functions which satisfy the following mean-value properties:(0.3) f(x)=jj(x + ty)dp(y), Presented to the Society,