A note on classical and p-adic Fr\'{e}chet functional equations with restrictions

TL;DR

This paper investigates conditions under which solutions to certain Fréchet functional equations, constrained by specific sets in vector spaces, extend from partial to full domain solutions, with focus on classical and p-adic contexts.

Contribution

It provides new criteria for extending solutions of Fréchet functional equations from restricted sets to entire spaces in both classical and p-adic frameworks.

Findings

Derived conditions for solution extension in classical vector spaces.

Established analogous results in p-adic vector spaces.

Clarified the role of set restrictions in functional equation solutions.

Abstract

Given X,Y two Q-vector spaces, and f:X -> Y, we study under which conditions on the sets $B_k\subseteq X$, k=1,...,s, if $\Delta_{h_1h_2... h_s}f(x)=0$ for all x in X and h_k in B_k, k=1,2,...,s, then $\Delta_{h_1h_2... h_s}f(x)=0$ for all (x,h_1,...,h_s) in X^{s+1}.