A general functional equation and its stability

Suppose that V and B are vector spaces over Q, R or C and alpha(0), beta(0),..., alpha(m), beta(m) are scalar such that alpha(j) beta(k) - alpha(k) beta(j) not equal 0 whenever 0 <= j < k <= m. We prove that if f(k) : V --> B for 0 <= k <= m and (*) m Sigma k=0 fk(alpha(k)x+ beta(k)y) = 0 for all x, y is an element of V, then each f(k) is a "generalized" polynomial map of "degree" at most m-1. In case V = R-n 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.