Basic sets for some basic functional equations

Abstract

AbstractMotivated by the well known fact that any nonzero solution of the fundamental Cauchy functional equation may arbitrarily be prescribed on a Hamel basis, we deal with the following problem: given a functional equation $$\begin{aligned} E_1(\varphi ) = E_2(\varphi ) \end{aligned}$$ E 1 ( φ ) = E 2 ( φ ) with the unknown function $$\varphi : X \longrightarrow Y$$ φ : X ⟶ Y , what must the set $$\emptyset \ne Z \subset X$$ ∅ ≠ Z ⊂ X be like in order to ensure that an arbitrary function $$\varphi _0: Z \longrightarrow Y$$ φ 0 : Z ⟶ Y admits a unique function $$\varphi : X \longrightarrow Y$$ φ : X ⟶ Y solving equation ($$*$$ ∗ ) and such that $$\varphi _{|_Z} = \varphi _0$$ φ | Z = φ 0 ; if such a set does exist it is termed to be a basic set. We discuss the problem of existence of basic sets for exponential functions, d’Alembert’s functions, sine functions, Cuculière’s functions and hyperbolic tangent type functions, among others.