Basic sets for some basic functional equations (Q6185274)

Let \(X\) and \(Y\) be linear spaces over the rationals and let \(H\) be a Hamel base of \(X\). By a well-known result of \textit{G. Hamel} [Math. Ann. 60, 459--462 (1905; JFM 36.0446.04)], any mapping \(f_{0}\colon H\to Y\) may uniquely be extended to a solution \(f \colon X \to Y\) of the fundamental Cauchy functional equation \[ f (x + y) = f (x) + f (y). \] Motivated by this fact and following [\textit{M. Sablik}, Basic sets for functional equations. Katowice: Wydawnictwo Uniwersytetu Śląskiego (1996; Zbl 0884.39008)], the author aims to study the following problem: given a functional equation \[ E_1(\varphi) = E_2 (\varphi)\tag{\(\ast\)} \] for the unknown function \(\varphi \colon X \to Y\), what is the set \(\emptyset \neq Z\subset X\) in order to ensure that an arbitrary function \(\varphi_{0}\colon Z \to Y\) admits a unique function \(\varphi \colon X \to Y\) solving equation \((\ast)\) and such that \(\varphi\vert_{Z}= \varphi_{0}\)? If such a set does exist it is called a ``basic set''. The requirement of uniqueness guarantees here that a basic set is the largest possible in the sense of inclusion. The author studies here the problem of existence of basic sets for exponential functions, d'Alembert's functions, sine functions, Cuculière's functions and hyperbolic tangent type functions.