Mergelyan's theorem is a result from approximation by polynomials in
complex analysis proved by the
Armenian
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* Something of, from, or related to Armenia, a country in the South Caucasus region of Eurasia
* Armenians, the national people of Armenia, or people of Armenian descent
** Armenian Diaspora, Armenian communities across the ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Sergei Mergelyan in 1951.
Statement
:Let ''K'' be a
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
of the
complex plane C such that C∖''K'' is
connected
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* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. Then, every
continuous function ''f'' : ''K''
C, such that the
restriction
Restriction, restrict or restrictor may refer to:
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* restrict, a keyword in the C programming language used in pointer declarations
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''f'' to int(''K'') is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
, can be approximated
uniformly on ''K'' with
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s. Here, int(''K'') denotes the
interior of ''K''.
Mergelyan's theorem also holds for open
Riemann surfaces
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
:If ''K'' is a compact set without holes in an open
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
''X'', then every function in
can be approximated uniformly on K by functions in
.
Mergelyan's theorem does not always hold in higher dimensions (spaces of
several complex variables), but it has some consequences.
History
Mergelyan's theorem is a generalization of the
Weierstrass approximation theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
and
Runge's theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:
Denoting by C the set of complex numbers, let '' ...
.
In the case that C∖''K'' is ''not'' connected, in the initial approximation problem the polynomials have to be replaced by
rational functions. An important step of the solution of this further
rational approximation problem was also suggested by Mergelyan in 1952. Further deep results on rational approximation are due to, in particular,
A. G. Vitushkin.
Weierstrass and Runge's theorems were put forward in 1885, while Mergelyan's theorem dates from 1951. After Weierstrass and Runge, many mathematicians (in particular
Walsh,
Keldysh,
Lavrentyev,
Hartogs, and
Rosenthal) had been working on the same problem. The method of the proof suggested by Mergelyan is constructive, and remains the only known constructive proof of the result.
See also
*
Arakelyan's theorem In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.
Theorem
Let Ω be an open subset of \Complex and ''E'' a re ...
*
Hartogs–Rosenthal theorem In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions. The theorem was proved in 1931 by the Germ ...
*
Oka–Weil theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil.
Statement
The Oka–Weil theorem ...
References
*
Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
, ''Mergelyan's theorem on uniform polynomial approximation'', Math. Scand., V. 15, (1964) 167–175.
* Dieter Gaier, ''Lectures on Complex Approximation'', Birkhäuser Boston, Inc. (1987), .
* W. Rudin, '' Real and Complex Analysis'', McGraw–Hill Book Co., New York, (1987), .
* A. G. Vitushkin, ''Half a century as one day'', Mathematical events of the twentieth century, 449–473, Springer, Berlin, (2006), /hbk.
Inline citation
External links
* {{springer, title=Mergelyan theorem, id=p/m063450
Theorems in complex analysis
Theorems in approximation theory