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 | + | <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) |
Montel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montel_theorem&oldid=52070