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Montel theorem

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Montel's theorem on the approximation of analytic functions by polynomials: If is an open set in the complex -plane not containing and is a single-valued function, analytic at each point , then there is a sequence of polynomials converging to at each . 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 be an infinite family of holomorphic functions in a domain of the complex -plane, then is pre-compact, that is, any subsequence has a subsequence converging uniformly on compact subsets of , if is uniformly bounded in . This theorem can be generalized to a domain in , (see Compactness principle).

Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [2]): Let be an infinite family of holomorphic functions in a domain of the complex -plane. If there are two distinct values and that are not taken by any of the functions , then is a normal family, that is, any sequence has a sequence uniformly converging on compact subsets of to a holomorphic function or to . The conditions of this theorem can be somewhat weakened: It suffices that all do not take one of the values, say , and that the other value is taken at most times, . This theorem can be generalized to a domain in , .

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. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Montel_theorem&oldid=19316
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098