Tauberian theorem

Wiener's tauberian theorem

In mathematics, Wiener's tauberian theorem is a 1932 result of Norbert Wiener. It put the capstone on the field of tauberian theorems 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 algebras, 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 combinations of translations f(xa) of f, with complex number coefficients, form a dense subspace in L1(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.

Search another word or see Tauberian theoremon Dictionary | Thesaurus |Spanish
Copyright © 2014 Dictionary.com, LLC. All rights reserved.
  • Please Login or Sign Up to use the Recent Searches feature