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.
Let f ∈ L1(R) be an integrable function. The span of translations fa(x) = f(x + a) is dense in L1(R) if and only if the Fourier transform of f has no real zeros.
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