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A theorem on the possibility of uniform approximation of functions of one complex variable by polynomials. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634501.png" /> be a compact subset of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634502.png" />-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634503.png" /> with a connected complement. Then every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634504.png" /> continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634505.png" /> and holomorphic at its interior points can be approximated uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634506.png" /> by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634507.png" />.
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A theorem on the possibility of uniform approximation of functions of one complex variable by polynomials. Let $K$ be a compact subset of the complex $z$-plane $\mathbf C$ with a connected complement. Then every function $f$ continuous on $K$ and holomorphic at its interior points can be approximated uniformly on $K$ by polynomials in $z$.
  
 
This theorem was proved by S.N. Mergelyan (see [[#References|[1]]], [[#References|[2]]]); it is the culmination of a large number of studies on approximation theory in the complex plane and has many applications in various branches of complex analysis.
 
This theorem was proved by S.N. Mergelyan (see [[#References|[1]]], [[#References|[2]]]); it is the culmination of a large number of studies on approximation theory in the complex plane and has many applications in various branches of complex analysis.
  
In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634508.png" /> has no interior points this result was proved by M.A. Lavrent'ev [[#References|[3]]]; the corresponding theorem in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m0634509.png" /> is a compact domain with a connected complement is due to M.V. Keldysh [[#References|[4]]] (cf. also [[Keldysh–Lavrent'ev theorem|Keldysh–Lavrent'ev theorem]]).
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In the case where $K$ has no interior points this result was proved by M.A. Lavrent'ev [[#References|[3]]]; the corresponding theorem in the case where $K$ is a compact domain with a connected complement is due to M.V. Keldysh [[#References|[4]]] (cf. also [[Keldysh–Lavrent'ev theorem|Keldysh–Lavrent'ev theorem]]).
  
Mergelyan's theorem has the following consequence. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345010.png" /> be an arbitrary compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345011.png" />. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345012.png" /> be continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345013.png" /> and holomorphic in its interior. Then in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345014.png" /> be uniformly approximable by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345015.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345016.png" /> admits a holomorphic extension to all bounded connected components of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345017.png" />.
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Mergelyan's theorem has the following consequence. Let $K$ be an arbitrary compact subset of $\mathbf C$. Let a function $f$ be continuous on $K$ and holomorphic in its interior. Then in order that $f$ be uniformly approximable by polynomials in $z$ it is necessary and sufficient that $f$ admits a holomorphic extension to all bounded connected components of the set $\mathbf C\setminus K$.
  
The problem of polynomial approximation is a particular case of the problem of approximation by rational functions with poles in the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345018.png" />. Mergelyan found also several sufficient conditions for rational approximation (see [[#References|[2]]]). A complete solution of this problem (for compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345019.png" />) was obtained in terms of analytic capacities (cf. [[Analytic capacity|Analytic capacity]]), [[#References|[5]]].
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The problem of polynomial approximation is a particular case of the problem of approximation by rational functions with poles in the complement of $K$. Mergelyan found also several sufficient conditions for rational approximation (see [[#References|[2]]]). A complete solution of this problem (for compacta $K\subset\mathbf C$) was obtained in terms of analytic capacities (cf. [[Analytic capacity|Analytic capacity]]), [[#References|[5]]].
  
Mergelyan's theorem touches upon a large number of papers concerning polynomial, rational and holomorphic approximation in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345020.png" /> of several complex variables. Here only partial results for special types of compact subsets have been obtained up till now.
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Mergelyan's theorem touches upon a large number of papers concerning polynomial, rational and holomorphic approximation in the space $\mathbf C^n$ of several complex variables. Here only partial results for special types of compact subsets have been obtained up till now.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Another important forerunner of Mergelyan's theorem was the Walsh theorem: the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345021.png" /> is the closure of a Jordan domain (a set with boundary consisting of Jordan curves, cf. [[Jordan curve|Jordan curve]]).
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Another important forerunner of Mergelyan's theorem was the Walsh theorem: the case where $K$ is the closure of a Jordan domain (a set with boundary consisting of Jordan curves, cf. [[Jordan curve|Jordan curve]]).
  
 
An interesting proof of Mergelyan's theorem, based on functional analysis, is due to L. Carlesson, see [[#References|[a1]]].
 
An interesting proof of Mergelyan's theorem, based on functional analysis, is due to L. Carlesson, see [[#References|[a1]]].
  
For analogues of Mergelyan's theorem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063450/m06345022.png" />, see [[#References|[a2]]]. See also [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]].
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For analogues of Mergelyan's theorem in $\mathbf C^n$, see [[#References|[a2]]]. See also [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Carlesson,  "Mergelyan's theorem on uniform polynomial approximation"  ''Math. Scand.'' , '''15'''  (1964)  pp. 167–175</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Chirka,  G.M. Khenkin,  "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5''' :  5  (1976)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  (1975)  pp. 13–142</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gaier,  "Lectures on complex approximation" , Birkhäuser  (1987)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Carlesson,  "Mergelyan's theorem on uniform polynomial approximation"  ''Math. Scand.'' , '''15'''  (1964)  pp. 167–175</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Chirka,  G.M. Khenkin,  "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5''' :  5  (1976)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  (1975)  pp. 13–142</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gaier,  "Lectures on complex approximation" , Birkhäuser  (1987)  (Translated from German)</TD></TR></table>

Latest revision as of 19:02, 1 May 2014

A theorem on the possibility of uniform approximation of functions of one complex variable by polynomials. Let $K$ be a compact subset of the complex $z$-plane $\mathbf C$ with a connected complement. Then every function $f$ continuous on $K$ and holomorphic at its interior points can be approximated uniformly on $K$ by polynomials in $z$.

This theorem was proved by S.N. Mergelyan (see [1], [2]); it is the culmination of a large number of studies on approximation theory in the complex plane and has many applications in various branches of complex analysis.

In the case where $K$ has no interior points this result was proved by M.A. Lavrent'ev [3]; the corresponding theorem in the case where $K$ is a compact domain with a connected complement is due to M.V. Keldysh [4] (cf. also Keldysh–Lavrent'ev theorem).

Mergelyan's theorem has the following consequence. Let $K$ be an arbitrary compact subset of $\mathbf C$. Let a function $f$ be continuous on $K$ and holomorphic in its interior. Then in order that $f$ be uniformly approximable by polynomials in $z$ it is necessary and sufficient that $f$ admits a holomorphic extension to all bounded connected components of the set $\mathbf C\setminus K$.

The problem of polynomial approximation is a particular case of the problem of approximation by rational functions with poles in the complement of $K$. Mergelyan found also several sufficient conditions for rational approximation (see [2]). A complete solution of this problem (for compacta $K\subset\mathbf C$) was obtained in terms of analytic capacities (cf. Analytic capacity), [5].

Mergelyan's theorem touches upon a large number of papers concerning polynomial, rational and holomorphic approximation in the space $\mathbf C^n$ of several complex variables. Here only partial results for special types of compact subsets have been obtained up till now.

References

[1] S.N. Mergelyan, "On the representation of functions by series of polynomials on closed sets" Transl. Amer. Math. Soc. , 3 (1962) pp. 287–293 Dokl. Akad. Nauk SSSR , 78 : 3 (1951) pp. 405–408
[2] S.N. Mergelyan, "Uniform approximation to functions of a complex variable" Transl. Amer. Math. Soc. , 3 (1962) pp. 294–391 Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 31–122
[3] M.A. [M.A. Lavrent'ev] Lavrentieff, "Sur les fonctions d'une variable complexe représentables par des series de polynômes" , Hermann (1936)
[4] M.V. Keldysh, "Sur la réprésentation par des séries de polynômes des fonctions d'une variable complexe dans des domaines fermés" Mat. Sb. , 16 : 3 (1945) pp. 249–258
[5] A.G. Vitushkin, "The analytic capacity of sets in problems of approximation theory" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199
[6] , Some questions in approximation theory , Moscow (1963) (In Russian; translated from English)
[7] T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969)


Comments

Another important forerunner of Mergelyan's theorem was the Walsh theorem: the case where $K$ is the closure of a Jordan domain (a set with boundary consisting of Jordan curves, cf. Jordan curve).

An interesting proof of Mergelyan's theorem, based on functional analysis, is due to L. Carlesson, see [a1].

For analogues of Mergelyan's theorem in $\mathbf C^n$, see [a2]. See also Approximation of functions of a complex variable.

References

[a1] L. Carlesson, "Mergelyan's theorem on uniform polynomial approximation" Math. Scand. , 15 (1964) pp. 167–175
[a2] E.M. Chirka, G.M. Khenkin, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 : 5 (1976) pp. 612–687 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 4 (1975) pp. 13–142
[a3] D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)
How to Cite This Entry:
Mergelyan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mergelyan_theorem&oldid=32115
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article