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- Mergelyan's_theorem abstract "Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Nikitovich Mergelyan in 1951. It states the following:Let K be a compact subset of the complex plane C such that C\K is connected. Then, every continuous function f : KC, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K.Mergelyan's theorem is the ultimate development and generalization of the Weierstrass approximation theorem and Runge's theorem. It gives the complete solution of the classical problem of approximation by polynomials.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. This rather large time difference is not surprising, as the proof of Mergelyan's theorem is based on a new powerful method created by Mergelyan. After Weierstrass and Runge, many mathematicians (in particular Walsh, Keldysh, and Lavrentyev) 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.".
- Mergelyan's_theorem wikiPageID "3277014".
- Mergelyan's_theorem wikiPageRevisionID "598462602".
- Mergelyan's_theorem hasPhotoCollection Mergelyan's_theorem.
- Mergelyan's_theorem id "p/m063450".
- Mergelyan's_theorem title "Mergelyan theorem".
- Mergelyan's_theorem subject Category:Theorems_in_approximation_theory.
- Mergelyan's_theorem subject Category:Theorems_in_complex_analysis.
- Mergelyan's_theorem type Abstraction100002137.
- Mergelyan's_theorem type Communication100033020.
- Mergelyan's_theorem type Message106598915.
- Mergelyan's_theorem type Proposition106750804.
- Mergelyan's_theorem type Statement106722453.
- Mergelyan's_theorem type Theorem106752293.
- Mergelyan's_theorem type TheoremsInApproximationTheory.
- Mergelyan's_theorem type TheoremsInComplexAnalysis.
- Mergelyan's_theorem comment "Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Nikitovich Mergelyan in 1951. It states the following:Let K be a compact subset of the complex plane C such that C\K is connected. Then, every continuous function f : KC, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials.".
- Mergelyan's_theorem label "Mergelyan's theorem".
- Mergelyan's_theorem label "Théorème de Mergelyan".
- Mergelyan's_theorem label "Теорема Мергеляна".
- Mergelyan's_theorem sameAs Théorème_de_Mergelyan.
- Mergelyan's_theorem sameAs m.092xkq.
- Mergelyan's_theorem sameAs Q1964537.
- Mergelyan's_theorem sameAs Q1964537.
- Mergelyan's_theorem sameAs Mergelyan's_theorem.
- Mergelyan's_theorem wasDerivedFrom Mergelyan's_theorem?oldid=598462602.
- Mergelyan's_theorem isPrimaryTopicOf Mergelyan's_theorem.