About: Wiener's Tauberian theorem

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In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L1 or L2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f can not approximate a function whose Fourier transform does not vanish on Z.

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rdfs:label	Théorème taubérien de Wiener (fr) Общая тауберова теорема Винера (ru) Wiener's Tauberian theorem (en)
rdfs:comment	En mathématiques, et plus précisément en analyse, le théorème taubérien de Wiener fait référence à plusieurs résultats analogues démontrés par Norbert Wiener en 1932. Ils donnent des conditions nécessaires et suffisantes pour pouvoir approximer une fonction des espaces L1 ou L2 par des combinaisons linéaires de translatées d'une fonction donnée. (fr) Общая тауберова теорема Винера — теорема об асимптотических свойствах линейных преобразований функций, имеющих не равное нулю преобразование Фурье. Была доказана Норбертом Винером в 1932 году. (ru) In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L1 or L2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f can not approximate a function whose Fourier transform does not vanish on Z. (en)
dcterms:subject	Harmonic analysis Tauberian theorems Real analysis
Wikipage page ID	4106793 (xsd:integer)
Wikipage revision ID	1083426666 (xsd:integer)
Link from a Wikipage to another Wikipage	Unit circle Convolution Harmonic analysis Mathematical analysis Mathematics Function (mathematics) Lp space Dense set Empty set Maximal ideal Lebesgue measure Linear combination Linear span Locally compact abelian group Fourier transform Banach algebra Norbert Wiener Mathematical proof Israel Gelfand Tauberian theorems Real analysis Zero of a function If and only if Real number Shift operator Wiener algebra Square-integrable function Zero set Tauberian theorem Integrable function Commutative C*-algebra
Link from a Wikipage to an external page	https://archive.org/details/functionalanalys00rudi
sameAs	Wiener's Tauberian theorem Wiener's Tauberian theorem Wiener's Tauberian theorem Wiener's Tauberian theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Citation_needed dbt:Math dbt:Reflist dbt:Harvs dbt:Eom
first	A.I. (en)
id	W/w097950 (en)
last	Shtern (en)
title	Wiener Tauberian theorem (en)
has abstract	En mathématiques, et plus précisément en analyse, le théorème taubérien de Wiener fait référence à plusieurs résultats analogues démontrés par Norbert Wiener en 1932. Ils donnent des conditions nécessaires et suffisantes pour pouvoir approximer une fonction des espaces L1 ou L2 par des combinaisons linéaires de translatées d'une fonction donnée. (fr) In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L1 or L2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f can not approximate a function whose Fourier transform does not vanish on Z. Wiener's theorems make this precise, stating that linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L1) or of Lebesgue measure zero (in the case of L2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L1 group ring L1(R) of the group R of real numbers is the dual group of R. A similar result is true when R is replaced by any locally compact abelian group. (en) Общая тауберова теорема Винера — теорема об асимптотических свойствах линейных преобразований функций, имеющих не равное нулю преобразование Фурье. Была доказана Норбертом Винером в 1932 году. (ru)
prov:wasDerivedFrom	wikipedia-en:Wiener's_Tauberian_theorem?oldid=1083426666&ns=0
page length (characters) of wiki page	6400 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Wiener's_Tauberian_theorem
is Link from a Wikipage to another Wikipage of	Wiener–Ikehara theorem Wiener's tauberian theorem Abelian and Tauberian theorems Littlewood's Tauberian theorem Wiener tauberian theorem
is Wikipage redirect of	Wiener's tauberian theorem Wiener tauberian theorem
is foaf:primaryTopic of	wikipedia-en:Wiener's_Tauberian_theorem

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