Montel theorem

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)

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)