Basic sets for some basic functional equations

Abstract

Motivated 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 *E1(φ)=E2(φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} E_1(\\varphi ) = E_2(\\varphi ) \\end{aligned}$$\\end{document}with the unknown function varphi : X longrightarrow Y, what must the set emptyset ne Z subset X be like in order to ensure that an arbitrary function varphi _0: Z longrightarrow Y admits a unique function varphi : X longrightarrow Y solving equation (*) and such that varphi _{|_Z} = varphi _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.