View source for Smirnov domain

''domain of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858402.png" />, domain of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858404.png" />''

A bounded simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858405.png" /> with a rectifiable Jordan boundary in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858406.png" /> having the following property: there is a univalent [[Conformal mapping|conformal mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858407.png" /> from the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858408.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s0858409.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584010.png" /> the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584011.png" /> can be written a the Poisson integral of its non-tangential boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584012.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584013.png" /></td> </tr></table>

These domains were introduced by V.I. Smirnov [[#References|[1]]] in 1928 in the course of investigating the completeness of a system of polynomials in the [[Smirnov class|Smirnov class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584014.png" />. The problem of the existence of non-Smirnov domains with rectifiable Jordan boundaries was solved by M.V. Keldysh and M.A. Lavrent'ev [[#References|[2]]], who gave a sophisticated and intricate construction of such domains and of the corresponding mapping functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584015.png" />, with the additional property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584016.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584017.png" />. The basic [[Boundary properties of analytic functions|boundary properties of analytic functions]] in the disc also hold for functions analytic in a Smirnov domain, and many of these properties hold only in Smirnov domains. Examples of Smirnov domains are Jordan domains whose boundaries are Lyapunov curves or piecewise Lyapunov curves with non-zero angles (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,   "Sur la théorie des polynomes orthogonaux à une variable complexe"  ''Zh. Leningrad. Fiz.-Mat. Obshch.'' , '''2''' :  1  (1928)  pp. 155–179</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,   M.A. Lavrent'ev,   "Sur la répresentation conforme des domaines limités"  ''Ann. Sci. École Normale Sup.'' , '''54'''  (1937)  pp. 1–38</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.J. Lohwater,   "The boundary behavior of analytic functions"  ''Itogi Nauk. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.Ts. Tumarkin,   "A sufficient condition for a domain to belong to class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584018.png" />"  ''Vestnik. Leningrad. Univ.'' , '''17''' :  13  (1962)  pp. 47–55  (In Russian)  (English summary)</TD></TR></table>



====Comments====
The German translation of Privalov's book is the most detailed Western reference on Smirnov classes and domains. A reference in English is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085840/s08584019.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>