A general functional equation and its stability

Abstract

Suppose that V and B are vector spaces over Q, R or C and α 0 , β 0 ,...,α m ,β m are scalar such that α j β k -α k β j ¬= 0 whenever 0 < j < k < m. We prove that if fk: V → B for 0 < k < m and formula math. then each f k is a generalized polynomial map of degree at most m - 1. In case V = R and B = C we show that if some f k is bounded on a set of positive inner Lebesgue measure, then it is a genuine polynomial function. Our main aim is to establish the stability of (*) (in the sense of Ulam) in case B is a Banach space. We also solve a distributional analogue of (*) and prove a mean value theorem concerning harmonic functions in two real variables.