A general functional equation and its stability

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

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.