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m (moved Schroeder functional equation to Schröder functional equation over redirect: accented title)
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The equation
 
The equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \phi ( f ( x ) ) = \lambda \phi ( x ), \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300702.png" /> is the unknown function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300703.png" /> is a known real-valued function of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300704.png" />. I.e. one asks for the eigenvalues and eigenfunctions of the composition operator (substitution operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300705.png" />. Sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300706.png" /> is allowed to be a function itself.
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300707.png" /></td> </tr></table>
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\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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300708.png" /></td> </tr></table>
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\begin{equation*} \phi ( \phi ( s , u ) , v ) = \phi ( s , u ^ { * } v ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300709.png" /></td> </tr></table>
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\begin{equation*} s \in S , u , v \in H , \phi : S \times H \rightarrow S, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007010.png" /> is a [[Semi-group|semi-group]], which asks for something like a right action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007012.png" />, [[#References|[a1]]], [[#References|[a4]]].
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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><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>
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<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=23520
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article