|
Mergelyan's Theorem
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951. Statement :Let ''K'' be a compact subset of the complex plane C such that C∖''K'' is connected. Then, every continuous function ''f'' : ''K''\to C, 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 also holds for open Riemann surfaces :If ''K'' is a compact set without holes in an open Riemann surface ''X'', then every function in \mathcal (K) can be approximated uniformly on K by functions in \mathcal(X). 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 and Runge's theorem. In the case that C∖''K'' is ''not'' co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for "his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems." Life He was a student of Arne Beurling and received his Ph.D. from Uppsala University in 1950. He did his post-doctoral work at Harvard University where he met and discussed Fourier series and their convergence with Antoni Zygmund and Raphaël Salem who were there in 1950 and 1951. He is a professor emeritus at Uppsala University, the Royal Institute of Technology in Stockholm, and the University of California, Los Angeles, and has served as director of the Mittag-Leffler Institute in Djursholm outside Stockholm 1968–1984. Between 1978 and 1982 he served as president of the International Mathematical Union. Carleson married Butte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 states that if ''X'' is a Stein space and ''K'' is a compact \mathcal(X)-convex subset of ''X'', then every holomorphic function in an open neighborhood of ''K'' can be approximated uniformly on ''K'' by holomorphic functions on \mathcal(X) (i.e. by polynomials). Applications Since Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The Behnke–Stein theorem was originally proved using the Oka–Weil theorem. See also * Oka coherence theorem References Bibliography * * * * * * Further reading * – An example where Runge's theorem In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 German mathematicians Friedrich Hartogs and Arthur Rosenthal and has been widely applied, particularly in operator theory. Statement The Hartogs–Rosenthal theorem states that if ''K'' is a compact subset of the complex plane with Lebesgue measure zero, then any continuous complex-valued function on ''K'' can be uniformly approximated by rational functions. Proof By the Stone–Weierstrass theorem any complex-valued continuous function on ''K'' can be uniformly approximated by a polynomial in z and \overline. So it suffices to show that \overline can be uniformly approximated by a rational function on ''K''. Let ''g(z)'' be a smooth function of compact support on C equal to 1 on ''K'' and set : f(z)=g(z)\cdot \overline. By the generaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 relatively closed subset of Ω. By Ω* is denoted the Alexandroff compactification of Ω. Arakelyan's theorem states that for every ''f'' continuous in ''E'' and holomorphic in the interior of ''E'' and for every ''ε'' > 0 there exists ''g'' holomorphic in Ω such that , ''g'' − ''f'', < ''ε'' on ''E'' if and only if Ω* \ ''E'' is connected and locally connected. See also * * |
Arthur Rosenthal
Arthur Rosenthal (24 February 1887, Fürth, Germany – 15 September 1959, Lafayette, Indiana) was a German mathematician. Career Rosenthal's mathematical studies started in 1905 in Munich, under Ferdinand Lindemann and Arnold Sommerfeld at the University of Munich and the Technical University Munich, as well as at the University of Göttingen. After submitting his thesis on regular polyhedra in 1909, he was promoted to assistant at the Technical University in 1911 and then associate professor in the University of Munich in 1920. The following year he was appointed associate professor in the University of Heidelberg, with a promotion to full professor in 1930. Between 1932 and 1933 he served as dean in the faculty of mathematics and natural sciences, but was forced from his university position as a result of Nazi policies against German Jews. He moved to the Netherlands in 1936 and from there emigrated to the United States in 1939. He was appointed lecturer and research fellow at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Hartogs
Friedrich Moritz "Fritz" Hartogs (20 May 1874 – 18 August 1943) was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables. Life Hartogs was the son of the merchant Gustav Hartogs and his wife Elise Feist and grew up in Frankfurt am Main. He studied at the Königliche Technische Hochschule Hannover, at the Technische Hochschule Charlottenburg, at the University of Berlin, and at the Ludwig Maximilian University of Munich, graduating with a doctorate in 1903 (supervised by Alfred Pringsheim). He did his Habilitation in 1905 and was Privatdozent and Professor in Munich (from 1910 to 1927 extraordinary professor and since 1927 ordinary professor). As a Jew, he suffered greatly under the Nazi regime: he was fired in 1935, was mistreated and briefly interned in KZ Dachau in 1938, and eventually committed suicide in 1943. Work Hartogs main work was in several complex variables where he is known for Hartogs's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Lavrentyev
Mikhail Alekseevich Lavrentyev (or Lavrentiev, russian: Михаи́л Алексе́евич Лавре́нтьев) (November 19, 1900 – October 15, 1980) was a Soviet Union, Soviet mathematician and hydrodynamics, hydrodynamicist. Early years Lavrentiev was born in Kazan, where his father was an instructor at a college (he later became a professor at Kazan University, then Moscow University). Lavrentiev entered Kazan University, and, when his family moved to Moscow in 1921, he transferred to the Department of Physics and Mathematics of Moscow University. He graduated in 1922. He continued his studies in the university in 1923-26 as a graduate student of Nikolai Luzin. Although Luzin was alleged to plagiarize in science and indulge in anti-Sovietism by some of his students in 1936, Lavrentiev did not participate in the notorious political persecution of his teacher which is known as the Nikolai Luzin#The Luzin affair of 1936, Luzin case or Nikolai Luzin#The Luzin affair of 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mstislav Keldysh
Mstislav Vsevolodovich Keldysh (russian: Мстисла́в Все́володович Ке́лдыш; – 24 June 1978) was a Soviet mathematician who worked as an engineer in the Soviet space program. He was the academician of the Academy of Sciences of the Soviet Union (1946), President of the Academy of Sciences of the Soviet Union (1961–1975), three-time Hero of Socialist Labour (1956, 1961, 1971), and fellow of the Royal Society of Edinburgh (1968). He was one of the key figures behind the Soviet space program. Among scientific circles of the USSR Keldysh was known by the epithet "the Chief Theoretician" in analogy with epithet "the Chief Designer" used for Sergei Korolev. Family Keldysh was born to a professional family of Russian nobility. His grandfather, Mikhail Fomich Keldysh (1839–1920), was a military physician, who retired with the military rank of General. Keldysh's grandmother, Natalia Keldysh (née Brusilova), was a cousin of general Aleksei Brusilov. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Leonard Walsh
__NOTOC__ Joseph Leonard Walsh (September 21, 1895 – December 6, 1973) was an American mathematician who worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szegő coincidence theorem is important in the study of the location of the zeros of multivariate polynomials. He became a member of the National Academy of Sciences in 1936 and served 1949–51 as president of the American Mathematical Society. Altogether he published 279 articles (research and others) and seven books, and advised 31 PhD students. For most of his professional career he studied and worked at Harvard University. He received a B.S. in 1916 and a PhD in 1920. The Advisor of his PhD was Maxime Bôcher. Walsh started to work as lecturer in Harvard afterwards and became a full professor in 1935. He was an Invited Speaker of the ICM in 1920 at Strasbourg. With two different scholarships he was able to study in Paris under Paul Montel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
