Smirnov class

From Encyclopedia of Mathematics
Revision as of 17:08, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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


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

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.


[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 spaces" , Acad. Press (1970)
How to Cite This Entry:
Smirnov class. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article