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Difference between revisions of "Smirnov domain"

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''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" />''
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{{TEX|done}}
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''domain of type $C$, domain of type $S$''
  
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" />:
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A bounded simply-connected domain $G$ with a rectifiable Jordan boundary in the complex plane $\mathbf C$ having the following property: there is a univalent [[Conformal mapping|conformal mapping]] $z=\phi(w)$ from the disc $|w|<1$ onto $G$ such that for $|w|<1$ the harmonic function $\ln|\phi'(w)|$ can be written a the Poisson integral of its non-tangential boundary values $\ln|\phi'(e^{i\theta})|$:
  
<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>
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$$\ln|\phi'(re^{i\theta})|=\frac{1}{2\pi}\int\limits_0^{2\pi}\frac{1-r^2}{1+r^2-2r\cos(t-\theta)}\ln|\phi'(e^{it})|dt.$$
  
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]]).
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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]] $E_2(G)$. 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 $\phi$, with the additional property that $|\phi'(e^{i\theta})|=1$ for almost-all $e^{i\theta}$. 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====
 
====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>
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<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 $S$"  ''Vestnik. Leningrad. Univ.'' , '''17''' :  13  (1962)  pp. 47–55  (In Russian)  (English summary)</TD></TR></table>
  
  
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====References====
 
====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>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Theory of $H^p$ spaces" , Acad. Press  (1970)</TD></TR></table>

Revision as of 17:57, 26 September 2014

domain of type $C$, domain of type $S$

A bounded simply-connected domain $G$ with a rectifiable Jordan boundary in the complex plane $\mathbf C$ having the following property: there is a univalent conformal mapping $z=\phi(w)$ from the disc $|w|<1$ onto $G$ such that for $|w|<1$ the harmonic function $\ln|\phi'(w)|$ can be written a the Poisson integral of its non-tangential boundary values $\ln|\phi'(e^{i\theta})|$:

$$\ln|\phi'(re^{i\theta})|=\frac{1}{2\pi}\int\limits_0^{2\pi}\frac{1-r^2}{1+r^2-2r\cos(t-\theta)}\ln|\phi'(e^{it})|dt.$$

These domains were introduced by V.I. Smirnov [1] in 1928 in the course of investigating the completeness of a system of polynomials in the Smirnov class $E_2(G)$. The problem of the existence of non-Smirnov domains with rectifiable Jordan boundaries was solved by M.V. Keldysh and M.A. Lavrent'ev [2], who gave a sophisticated and intricate construction of such domains and of the corresponding mapping functions $\phi$, with the additional property that $|\phi'(e^{i\theta})|=1$ for almost-all $e^{i\theta}$. The basic 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).

References

[1] V.I. Smirnov, "Sur la théorie des polynomes orthogonaux à une variable complexe" Zh. Leningrad. Fiz.-Mat. Obshch. , 2 : 1 (1928) pp. 155–179
[2] M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domaines limités" Ann. Sci. École Normale Sup. , 54 (1937) pp. 1–38
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] A.J. Lohwater, "The boundary behavior of analytic functions" Itogi Nauk. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)
[5] G.Ts. Tumarkin, "A sufficient condition for a domain to belong to class $S$" Vestnik. Leningrad. Univ. , 17 : 13 (1962) pp. 47–55 (In Russian) (English summary)


Comments

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

References

[a1] P.L. Duren, "Theory of $H^p$ spaces" , Acad. Press (1970)
How to Cite This Entry:
Smirnov domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smirnov_domain&oldid=13457
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article