Wiener's tauberian theorem 4106793 207239811 2008-04-21T23:40:36Z David Eppstein 2051880 stub sorting 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'' &minus; ''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. {{mathanalysis-stub}} [[Category:Real analysis]] [[Category:Harmonic analysis]] [[Category:Mathematical theorems]]