Wiener's tauberian theorem
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2008-04-21T23:40:36Z
David Eppstein
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In [[mathematics]], '''Wiener's tauberian theorem''' is a 1932 result of [[Norbert Wiener]]. It put the capstone on the field of [[tauberian theorem]]s in [[summability theory]], on the face of it a chapter of [[real analysis]], by showing that most of the known results could be encapsulated in a principle from [[harmonic analysis]]. As now formulated, the theorem of Wiener has no obvious connection to tauberian theorems, which deal with [[infinite series]]; the translation from results formulated for integrals, or using the language of [[functional analysis]] and [[Banach algebra]]s, is however a relatively routine process once the idea is grasped.
There are numerous statements that can be given. A simple abstract result is this: for an [[integrable function]] ''f(''x'')'' on the [[real line]] '''R''', such that the [[Fourier transform]] of ''f'' never takes the value 0, the finite [[linear combination]]s of translations ''f''(''x'' − ''a'') of ''f'', with complex number coefficients, form a [[dense subspace]] in ''L''<sup>1</sup>('''R'''). (This is given, for example, in K. Yoshida, ''Functional Analysis''.)
==Further reading==
Norbert Wiener, "Tauberian theorem", Annals of Mathematics '''33''' (1932), pp. 1–100.
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[[Category:Real analysis]]
[[Category:Harmonic analysis]]
[[Category:Mathematical theorems]]