Namespaces
Variants
Actions

Difference between revisions of "Montel theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
Montel's theorem on the approximation of analytic functions by polynomials: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648901.png" /> is an open set in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648902.png" />-plane not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648904.png" /> is a single-valued function, analytic at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648905.png" />, then there is a sequence of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648906.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648907.png" /> at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648908.png" />. This theorem is one of the basic results in the theory of [[Approximation of functions of a complex variable|approximation of functions of a complex variable]]; it was obtained by P. Montel .
+
<!--
 +
m0648901.png
 +
$#A+1 = 37 n = 0
 +
$#C+1 = 37 : ~/encyclopedia/old_files/data/M064/M.0604890 Montel theorem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Montel's theorem on compactness conditions for a family of holomorphic functions (principle of compactness, see ): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m0648909.png" /> be an infinite family of holomorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489010.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489011.png" />-plane, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489012.png" /> is pre-compact, that is, any subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489013.png" /> has a subsequence converging uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489014.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489015.png" /> is uniformly bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489016.png" />. This theorem can be generalized to a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489019.png" /> (see [[Compactness principle|Compactness principle]]).
+
{{TEX|auto}}
 +
{{TEX|done}}
  
Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [[#References|[2]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489020.png" /> be an infinite family of holomorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489021.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489022.png" />-plane. If there are two distinct values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489024.png" /> that are not taken by any of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489026.png" /> is a [[Normal family|normal family]], that is, any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489027.png" /> has a sequence uniformly converging on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489028.png" /> to a holomorphic function or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489029.png" />. The conditions of this theorem can be somewhat weakened: It suffices that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489030.png" /> do not take one of the values, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489031.png" />, and that the other value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489032.png" /> is taken at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489033.png" /> times, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489034.png" />. This theorem can be generalized to a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064890/m06489037.png" />.
+
Montel's theorem on the approximation of analytic functions by polynomials: If  $  D $
 +
is an open set in the complex  $  z $-
 +
plane not containing  $  z = \infty $
 +
and  $  f ( z) $
 +
is a single-valued function, analytic at each point  $  z \in D $,
 +
then there is a sequence of polynomials  $  \{ P _ {n} ( z) \} $
 +
converging to  $  f ( z) $
 +
at each  $  z \in D $.
 +
This theorem is one of the basic results in the theory of [[Approximation of functions of a complex variable|approximation of functions of a complex variable]]; it was obtained by P. Montel .
 +
 
 +
Montel's theorem on compactness conditions for a family of holomorphic functions (principle of compactness, see ): Let  $  \Phi = \{ f ( z) \} $
 +
be an infinite family of holomorphic functions in a domain  $  D $
 +
of the complex  $  z $-
 +
plane, then  $  \Phi $
 +
is pre-compact, that is, any subsequence  $  \{ f _ {k} ( z) \} \subset  \Phi $
 +
has a subsequence converging uniformly on compact subsets of  $  D $,
 +
if  $  \Phi $
 +
is uniformly bounded in  $  D $.
 +
This theorem can be generalized to a domain  $  D $
 +
in  $  \mathbf C  ^ {n} $,
 +
$  n \geq  1 $(
 +
see [[Compactness principle|Compactness principle]]).
 +
 
 +
Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [[#References|[2]]]): Let $  \Phi = \{ f ( z) \} $
 +
be an infinite family of holomorphic functions in a domain $  D $
 +
of the complex $  z $-
 +
plane. If there are two distinct values $  a $
 +
and $  b $
 +
that are not taken by any of the functions $  f ( z) \in \Phi $,  
 +
then $  \Phi $
 +
is a [[Normal family|normal family]], that is, any sequence $  \{ f _ {k} ( z) \} \subset  \Phi $
 +
has a sequence uniformly converging on compact subsets of $  D $
 +
to a holomorphic function or to $  \infty $.  
 +
The conditions of this theorem can be somewhat weakened: It suffices that all $  f ( z) \in \Phi $
 +
do not take one of the values, say $  a $,  
 +
and that the other value $  b $
 +
is taken at most m $
 +
times, $  1 \leq  m < \infty $.  
 +
This theorem can be generalized to a domain $  D $
 +
in $  \mathbf C  ^ {n} $,  
 +
$  n \geq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Montel,  "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars  (1910)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Montel,  "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars  (1910)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3, Sect. 11; 1, Sect. 86; 3, Sect. 50''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3, Sect. 11; 1, Sect. 86; 3, Sect. 50''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>

Revision as of 08:01, 6 June 2020


Montel's theorem on the approximation of analytic functions by polynomials: If $ D $ is an open set in the complex $ z $- plane not containing $ z = \infty $ and $ f ( z) $ is a single-valued function, analytic at each point $ z \in D $, then there is a sequence of polynomials $ \{ P _ {n} ( z) \} $ converging to $ f ( z) $ at each $ z \in D $. This theorem is one of the basic results in the theory of approximation of functions of a complex variable; it was obtained by P. Montel .

Montel's theorem on compactness conditions for a family of holomorphic functions (principle of compactness, see ): Let $ \Phi = \{ f ( z) \} $ be an infinite family of holomorphic functions in a domain $ D $ of the complex $ z $- plane, then $ \Phi $ is pre-compact, that is, any subsequence $ \{ f _ {k} ( z) \} \subset \Phi $ has a subsequence converging uniformly on compact subsets of $ D $, if $ \Phi $ is uniformly bounded in $ D $. This theorem can be generalized to a domain $ D $ in $ \mathbf C ^ {n} $, $ n \geq 1 $( see Compactness principle).

Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [2]): Let $ \Phi = \{ f ( z) \} $ be an infinite family of holomorphic functions in a domain $ D $ of the complex $ z $- plane. If there are two distinct values $ a $ and $ b $ that are not taken by any of the functions $ f ( z) \in \Phi $, then $ \Phi $ is a normal family, that is, any sequence $ \{ f _ {k} ( z) \} \subset \Phi $ has a sequence uniformly converging on compact subsets of $ D $ to a holomorphic function or to $ \infty $. The conditions of this theorem can be somewhat weakened: It suffices that all $ f ( z) \in \Phi $ do not take one of the values, say $ a $, and that the other value $ b $ is taken at most $ m $ times, $ 1 \leq m < \infty $. This theorem can be generalized to a domain $ D $ in $ \mathbf C ^ {n} $, $ n \geq 1 $.

References

[1] P. Montel, "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars (1910)
[2] P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927)

Comments

References

[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 3, Sect. 11; 1, Sect. 86; 3, Sect. 50 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Montel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montel_theorem&oldid=47897
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article