Difference between revisions of "Schröder functional equation"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (moved Schröder functional equation to Schroeder functional equation: ascii title) |
m (AUTOMATIC EDIT (latexlist): Replaced 12 formulas out of 12 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, | ||
| + | the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist | ||
| + | was used. | ||
| + | If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. | ||
| + | |||
| + | Out of 12 formulas, 12 were replaced by TEX code.--> | ||
| + | |||
| + | {{TEX|semi-auto}}{{TEX|done}} | ||
The equation | The equation | ||
| − | + | \begin{equation} \tag{a1} \phi ( f ( x ) ) = \lambda \phi ( x ), \end{equation} | |
| − | where | + | 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 | 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|Functional equation]]) have much to do with the translation functional equation | The Schröder and Abel functional equations (see also [[Functional equation|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 | + | where $H$ is a [[Semi-group|semi-group]], which asks for something like a right action of $H$ on $S$, [[#References|[a1]]], [[#References|[a4]]]. |
The equation was formulated by E. Schröder, [[#References|[a5]]], and there is an extensive body of literature. | The equation was formulated by E. Schröder, [[#References|[a5]]], and there is an extensive body of literature. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> J. Aczél, "A short course on functional equations" , Reidel (1987)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Kuczma, "On the Schröder operator" , PWN (1963)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> M. Kuczma, "Functional equations in a single variable" , PWN (1968)</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> G. Targonski, "Topics in iteration theory" , Vandenhoeck and Ruprecht (1981) pp. 82ff.</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> E. Schröder, "Uber iterierte Funktionen III" ''Math. Ann.'' , '''3''' (1970) pp. 296–322</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> J. Walorski, "Convex solutions of the Schröder equation in Banach spaces" ''Proc. Amer. Math. Soc.'' , '''125''' (1997) pp. 153–158</td></tr></table> |
Latest revision as of 16:58, 1 July 2020
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.
References
| [a1] | J. Aczél, "A short course on functional equations" , Reidel (1987) |
| [a2] | M. Kuczma, "On the Schröder operator" , PWN (1963) |
| [a3] | M. Kuczma, "Functional equations in a single variable" , PWN (1968) |
| [a4] | G. Targonski, "Topics in iteration theory" , Vandenhoeck and Ruprecht (1981) pp. 82ff. |
| [a5] | E. Schröder, "Uber iterierte Funktionen III" Math. Ann. , 3 (1970) pp. 296–322 |
| [a6] | J. Walorski, "Convex solutions of the Schröder equation in Banach spaces" Proc. Amer. Math. Soc. , 125 (1997) pp. 153–158 |
How to Cite This Entry:
Schröder functional equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schr%C3%B6der_functional_equation&oldid=23007
Schröder functional equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schr%C3%B6der_functional_equation&oldid=23007
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article