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.

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