# Schröder functional equation

The equation

\begin{equation} \tag{a1} \phi ( f ( x ) ) = \lambda \phi ( x ), \end{equation}

where $\phi$ is the unknown function and $f ( x )$ is a known real-valued function of a real variable $x$. I.e. one asks for the eigenvalues and eigenfunctions of the composition operator (substitution operator) $\phi \mapsto \phi \circ f$. Sometimes $\lambda$ is allowed to be a function itself.

One also considers the non-autonomous Schröder functional equation

\begin{equation*} \phi ( f ( x ) ) = g ( x ) \phi ( x ) + h ( x ). \end{equation*}

The Schröder and Abel functional equations (see also Functional equation) have much to do with the translation functional equation

\begin{equation*} \phi ( \phi ( s , u ) , v ) = \phi ( s , u ^ { * } v ), \end{equation*}

\begin{equation*} s \in S , u , v \in H , \phi : S \times H \rightarrow S, \end{equation*}

where $H$ is a semi-group, which asks for something like a right action of $H$ on $S$, [a1], [a4].

The equation was formulated by E. Schröder, [a5], and there is an extensive body of literature.

How to Cite This Entry:
Schröder functional equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schr%C3%B6der_functional_equation&oldid=50251
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article