Real analysis | Harmonic analysis | Tauberian theorems

Wiener's Tauberian theorem

In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L1 or L2 can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f vanishes on a certain set Z, the Fourier transform of any linear combination of translations of f also vanishes on Z. Therefore, the linear combinations of translations of f can not approximate a function whose Fourier transform does not vanish on Z. Wiener's theorems make this precise, stating that linear combinations of translations of f are dense if and only if the zero set of the Fourier transform of f is empty (in the case of L1) or of Lebesgue measure zero (in the case of L2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L1 group ring L1(R) of the group R of real numbers is the dual group of R. A similar result is true when R is replaced by any locally compact abelian group. (Wikipedia).

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From playlist Manin conjectures and rational points

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From playlist Algebraic geometry: extra topics

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From playlist Programming

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From playlist A Mathematical Tribute to Ennio De Giorgi

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From playlist Bourbaphy - 17/11/18 - L'information

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Locally compact abelian group | Lebesgue measure | Linear span | If and only if | Israel Gelfand | Zero of a function | Wiener algebra | Maximal ideal | Mathematical analysis | Mathematical proof | Banach algebra | Empty set | Square-integrable function | Dense set | Mathematics | Unit circle | Function (mathematics) | Real number | Linear combination | Convolution | Shift operator | Lp space | Fourier transform