A general functional equation and its stability

Abstract:

Suppose that $V$ and $B$ are vector spaces over $\mathbb {Q},\ \mathbb {R}$ or $\mathbb {C}$ and $\alpha _0,\beta _0,\dots ,\alpha _m,\beta _m$ are scalar such that $\alpha _j\beta _k-\alpha _k\beta _j\neq 0$ whenever $0\leq j<k\leq m.$ We prove that if $f_k:V\rightarrow B$ for $0\leq k\leq m$ and \begin{equation*} (*)\quad \qquad \qquad \qquad \sum ^m_{k=0} f_k(\alpha _kx+\beta _ky)=0\quad \text {for all}\ x,y\in V, \qquad \quad \qquad \qquad \quad \end{equation*} then each $f_k$ is a “generalized” polynomial map of “degree” at most $m-1.$ In case $V=\mathbb {R}^n$ and $B=\mathbb {C}$ 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.