Abelian and Tauberian theorems (original) (raw)
En mathématiques, et plus précisément en analyse, on appelle théorèmes abéliens et taubériens des théorèmes donnant des conditions pour que des méthodes distinctes de sommation de séries aboutissent au même résultat. Leurs noms viennent de Niels Henrik Abel et Alfred Tauber, les premiers exemples en étant le théorème d'Abel montrant que la sommation d'Abel d'une série convergente a pour valeur la somme de cette série, et le théorème de Tauber montrant que si la sommation d'Abel est possible, et que les coefficients de la série considérée sont suffisamment petits, alors la série converge (vers sa somme d'Abel).
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| dbo:abstract | In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In the theory of integral transforms, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function. (en) En mathématiques, et plus précisément en analyse, on appelle théorèmes abéliens et taubériens des théorèmes donnant des conditions pour que des méthodes distinctes de sommation de séries aboutissent au même résultat. Leurs noms viennent de Niels Henrik Abel et Alfred Tauber, les premiers exemples en étant le théorème d'Abel montrant que la sommation d'Abel d'une série convergente a pour valeur la somme de cette série, et le théorème de Tauber montrant que si la sommation d'Abel est possible, et que les coefficients de la série considérée sont suffisamment petits, alors la série converge (vers sa somme d'Abel). (fr) 타우버의 정리(Tauber's theorem, -定理)는 해석학의 초등적인 정리 중 하나로, 오스트리아-헝가리 제국의 수학자 (Alfred Tauber)의 이름이 붙어 있다. 슈톨츠-체사로 정리의 부분적 역을 제공하는 정리이다. (ko) Em matemática, teoremas abeliano e tauberiano relacionam-se à atribuição de um valor significativo como a "soma" da classe de séries divergentes. Um grande número de métodos tem sido propostos para o somatório de tais séries, geralmente tomando a forma de algum funcional linear L com domínio contido em algum espaço S de sequências numéricas. Este é, primeiramente, um método útil para atribuir uma soma a uma série que não convergiria a ser linear. Secundariamente, a sequência de somas parciais das séries é considerada, a qual é uma maneira equivalente de apresentá-la. (pt) Теорема Абеля — Таубера — теорема, обратная теореме Абеля о степенных рядах. Первая теорема типа тауберовых теорем. Была доказана A. Таубером в 1897 г. (теорема Таубера) Формулировку и доказательство при более общих условиях затем дал Дж. Литтльвуд в 1910 г. Затем была доказана Р. Шмидтом, Н. Винером. Наиболее простое доказательство дал Дж. Карамата. Формулировку и доказательство при более слабом условии дал Э. Ландау. (ru) |
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| dbp:authorlink | Alfred Tauber (en) |
| dbp:first | Alfred (en) |
| dbp:id | T/t092280 (en) |
| dbp:last | Tauber (en) |
| dbp:title | Tauberian theorems (en) |
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| rdfs:comment | En mathématiques, et plus précisément en analyse, on appelle théorèmes abéliens et taubériens des théorèmes donnant des conditions pour que des méthodes distinctes de sommation de séries aboutissent au même résultat. Leurs noms viennent de Niels Henrik Abel et Alfred Tauber, les premiers exemples en étant le théorème d'Abel montrant que la sommation d'Abel d'une série convergente a pour valeur la somme de cette série, et le théorème de Tauber montrant que si la sommation d'Abel est possible, et que les coefficients de la série considérée sont suffisamment petits, alors la série converge (vers sa somme d'Abel). (fr) 타우버의 정리(Tauber's theorem, -定理)는 해석학의 초등적인 정리 중 하나로, 오스트리아-헝가리 제국의 수학자 (Alfred Tauber)의 이름이 붙어 있다. 슈톨츠-체사로 정리의 부분적 역을 제공하는 정리이다. (ko) Em matemática, teoremas abeliano e tauberiano relacionam-se à atribuição de um valor significativo como a "soma" da classe de séries divergentes. Um grande número de métodos tem sido propostos para o somatório de tais séries, geralmente tomando a forma de algum funcional linear L com domínio contido em algum espaço S de sequências numéricas. Este é, primeiramente, um método útil para atribuir uma soma a uma série que não convergiria a ser linear. Secundariamente, a sequência de somas parciais das séries é considerada, a qual é uma maneira equivalente de apresentá-la. (pt) Теорема Абеля — Таубера — теорема, обратная теореме Абеля о степенных рядах. Первая теорема типа тауберовых теорем. Была доказана A. Таубером в 1897 г. (теорема Таубера) Формулировку и доказательство при более общих условиях затем дал Дж. Литтльвуд в 1910 г. Затем была доказана Р. Шмидтом, Н. Винером. Наиболее простое доказательство дал Дж. Карамата. Формулировку и доказательство при более слабом условии дал Э. Ландау. (ru) In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. (en) |
| rdfs:label | Abelian and Tauberian theorems (en) Théorèmes abéliens et taubériens (fr) 타우버의 정리 (ko) Teoremas abeliano e tauberiano (pt) Теорема Абеля — Таубера (ru) |
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