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Difference between revisions of "Montel theorem"

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Montel's theorem on the approximation of analytic functions by polynomials: If  $  D $
 
Montel's theorem on the approximation of analytic functions by polynomials: If  $  D $
is an open set in the complex  $  z $-
+
is an open set in the complex  $  z $-plane not containing  $  z = \infty $
plane not containing  $  z = \infty $
 
 
and  $  f ( z) $
 
and  $  f ( z) $
 
is a single-valued function, analytic at each point  $  z \in D $,  
 
is a single-valued function, analytic at each point  $  z \in D $,  
Line 23: Line 22:
 
Montel's theorem on compactness conditions for a family of holomorphic functions (principle of compactness, see ): Let  $  \Phi = \{ f ( z) \} $
 
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 $
 
be an infinite family of holomorphic functions in a domain  $  D $
of the complex  $  z $-
+
of the complex  $  z $-plane, then  $  \Phi $
plane, then  $  \Phi $
 
 
is pre-compact, that is, any subsequence  $  \{ f _ {k} ( z) \} \subset  \Phi $
 
is pre-compact, that is, any subsequence  $  \{ f _ {k} ( z) \} \subset  \Phi $
 
has a subsequence converging uniformly on compact subsets of  $  D $,  
 
has a subsequence converging uniformly on compact subsets of  $  D $,  
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This theorem can be generalized to a domain  $  D $
 
This theorem can be generalized to a domain  $  D $
 
in  $  \mathbf C  ^ {n} $,  
 
in  $  \mathbf C  ^ {n} $,  
$  n \geq  1 $(
+
$  n \geq  1 $ (see [[Compactness principle|Compactness principle]]).
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) \} $
 
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 $
 
be an infinite family of holomorphic functions in a domain  $  D $
of the complex  $  z $-
+
of the complex  $  z $-plane. If there are two distinct values  $  a $
plane. If there are two distinct values  $  a $
 
 
and  $  b $
 
and  $  b $
 
that are not taken by any of the functions  $  f ( z) \in \Phi $,  
 
that are not taken by any of the functions  $  f ( z) \in \Phi $,  
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====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>

Latest revision as of 12:08, 18 February 2022


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=52070
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article