Smirnov class
The set of all functions
holomorphic in a simply-connected domain
with rectifiable Jordan boundary
, such that for every function in it there is a sequence of closed rectifiable Jordan curves
,
with the following properties:
1) tends to
as
in the sense that if
is the bounded domain with boundary
, then
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2)
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This definition was proposed by M.V. Keldysh and M.A. Lavrent'ev [2], and is equivalent to V.I. Smirnov's definition [1] in which curves are used instead of
. These curves are the images of the circles
under some univalent conformal mapping
from the disc
onto the domain
, and the supremum is taken over all
.
The classes are the best known and most thoroughly studied generalization of the Hardy classes
, and are connected with them by the following relation:
if and only if
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The properties of the classes are closest to those of
in the case when
is a Smirnov domain. They have been generalized to domains
with boundaries of finite Hausdorff length. See also Boundary properties of analytic functions.
References
[1] | V.I. Smirnov, "Sur les formules de Cauchy et de Green et quelques problèmes qui s'y rattachent" Izv. Akad. Nauk SSSR. Otdel. Mat. i Estestv. Nauk , 3 (1932) pp. 337–372 |
[2] | M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domaines limités par des courbes rectifiables" Ann. Sci. Ecole Norm. Sup. , 54 (1937) pp. 1–38 |
[3] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[4] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[5] | P.L. Duren, "Theory of ![]() |
Smirnov class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smirnov_class&oldid=48738