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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was a U.S. mathematician and applied mathematician, especially in the field of electronics engineering. ... In mathematics, a large number of methods have been proposed for the summation of divergent series. ... In mathematics, a divergent series is a series that does not converge. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, a series is a sum of a sequence of terms. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

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 translates f(x − a) of f, with complex number coefficients, form a dense subspace in L¹(R). (This is given, for example, in K. Yoshida, Functional Analysis.) In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, the real line is simply the set of real numbers. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...