Montel theorem
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) |
Montel theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montel_theorem&oldid=19316