Difference between revisions of "Mergelyan theorem"
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− | A theorem on the possibility of uniform approximation of functions of one complex variable by polynomials. Let | + | {{TEX|done}} |
+ | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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) |
Mergelyan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mergelyan_theorem&oldid=32115