Wiener's_Tauberian_theorem
In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.[1] They provide a necessary and sufficient condition under which any function in or can be approximated by linear combinations of translations of a given function.[2]
Informally, if the Fourier transform of a function vanishes on a certain set , the Fourier transform of any linear combination of translations of also vanishes on . Therefore, the linear combinations of translations of cannot approximate a function whose Fourier transform does not vanish on .
Wiener's theorems make this precise, stating that linear combinations of translations of are dense if and only if the zero set of the Fourier transform of is empty (in the case of ) or of Lebesgue measure zero (in the case of ).
Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the group ring of the group of real numbers is the dual group of . A similar result is true when is replaced by any locally compact abelian group.