We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{ Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{ h_0}^{ m+1}f(x)=0 \ \ \text{ for all} x\in\mathbb{ Q}_p, \] and $|h_0|_p=p^{ -N_0}$ then, for all $x_0\in \mathbb{ Q}_p$, the restriction of $f$ over the set $x_0+p^{ N_0}\mathbb{ Z}_p$ coincides with a polynomial $p_{ x_0}(x)=a_0(x_0)+a_1(x_0)x+...+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by { equation} \Delta_h^{ m+1}f(x)=0 \ \ (x\in X \text{ and} h\in B_X(r)=\{ x\in X:\|x\|\leq r\}), { equation} whenever $f:X\to Y$, $X$ is an ultrametric normed space over a non-Archimedean valued field $(\mathbb{ K},|...|)$ of characteristic zero, and $Y$ is a $\mathbb{ Q}$-vector space. By obvious reasons, we call these functions uniformly locally polynomial.
Categories: math.CA