A Note on Classical and p-adic Fréchet Functional Equations with Restrictions

Abstract

Given X,Y two ${\mathbb{Q}}$-vector spaces, and f : X → Y, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$, if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$, then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$.