On extending functions to solutions of some functional equations
This is a paper related to extension of functions satisfying Cauchy and D'Alembert functional equations \((1)\quad g(x+y)=g(x)g(y)\) and \((2)\quad F(x+y)+F(x-y)=2F(x)F(y).\) It is a partial answer to a broader question raised by R. Ger: Given a functional equation (E) \(A_ 1=A_ 2\), where \(A_ i\), \(i=1,2\), are terms containing an unknown function \(\phi\), which is supposed to be defined on a set X, is it possible to find such a set \(Z\subset X\) that (I) every function \(\psi\) defined on Z has a unique extension to a solution of (E)? For example regarding (2) which was dealt by the reviewer, Vietoris and others, the author proves the following result. Let H be a Hamel basis of a linear space over rationals E. Let \(Z=\cup_{p\in P}\cup_{k\in N_ 0}p^{-k}H\) and let \(F_ 0:Z\to {\mathbb{C}}\) be an arbitrary function. (i) If \(F_ 0=1\) then \(F=1\) is its unique extension to a solution of (2). (ii) Let \(F_ 0\neq 1\). If \(F_ 0\) has an extension to a solution of (2) then \(F_ 0=(g_ 0+1/g_ 0)\) where \(g_ 0:Z\to {\mathbb{C}}^*\) is a function satisfying \(g_ 0(p^{- k}h)=g_ 0(p^{-k+1}h)\) for all \(p\in {\mathbb{P}}\), \(k\in {\mathbb{N}}\) and \(h\in H\). Conversely, if such a function \(g_ 0\) exists then \(F_ 0(z_ 0)^ 2\neq 1\) for a \(z_ 0\in Z\) and the function \(F_ 0:Z\cup (Z+z_ 0)\to {\mathbb{C}}\) given by \[ F_ 0(z)= \begin{cases} F_ 0(z),& z\in Z,\\ (g_ 0(z-z_ 0)g_ 0(z_ 0)+g_ 0(z-z_ 0)^{-1}g_ 0(z_ 0)^{-1}),& z\in Z+z_ 0 \end{cases} \] has a unique extension to a solution to (2).
scientific article; zbMATH DE number 3975504