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 L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can be approximated by linear combinations of translations of a given function. Informally, if the Fourier transform of a function f {\displaystyle f} vanishes on a certain set Z {\displaystyle Z} , the Fourier transform of any linear combination of translations of f {\displaystyle f} also vanishes on Z {\displaystyle Z} . Therefore, the linear combinations of translations of f {\displaystyle f} cannot approximate a function whose Fourier transform does not vanish on Z {\displaystyle Z} . Wiener's theorems make this precise, stating that linear combinations of translations of f {\displaystyle f} are dense if and only if the zero set of the Fourier transform of f {\displaystyle f} is empty (in the case of L 1 {\displaystyle L^{1}} ) or of Lebesgue measure zero (in the case of L 2 {\displaystyle L^{2}} ). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L 1 {\displaystyle L^{1}} group ring
L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} of the group R {\displaystyle \mathbb {R} } of real numbers is the dual group of R {\displaystyle \mathbb {R} } . A similar result is true when
R {\displaystyle \mathbb {R} } is replaced by any locally compact abelian group.
Introduction A typical Tauberian theorem is the following result, for f ∈ L 1 ( 0 , ∞ ) {\displaystyle f\in L^{1}(0,\infty )} . If:
f ( x ) = O ( 1 ) {\displaystyle f(x)=O(1)} as x → ∞ {\displaystyle x\to \infty }
1 x ∫ 0 ∞ e − t / x f ( t ) d t → L {\displaystyle {\frac {1}{x}}\int _{0}^{\infty }e^{-t/x}f(t)\,dt\to L} as x → ∞ {\displaystyle x\to \infty } , then
1 x ∫ 0 x f ( t ) d t → L . {\displaystyle {\frac {1}{x}}\int _{0}^{x}f(t)\,dt\to L.}
Generalizing, let G ( t ) {\displaystyle G(t)} be a given function, and P G ( f ) {\displaystyle P_{G}(f)} be the proposition
1 x ∫ 0 ∞ G ( t / x ) f ( t ) d t → L . {\displaystyle {\frac {1}{x}}\int _{0}^{\infty }G(t/x)f(t)\,dt\to L.}
Note that one of the hypotheses and the conclusion of the Tauberian theorem has the form P G ( f ) {\displaystyle P_{G}(f)} , respectively, with G ( t ) = e − t {\displaystyle G(t)=e^{-t}} and G ( t ) = 1 [ 0 , 1 ] ( t ) . {\displaystyle G(t)=1_{[0,1]}(t).}
The second hypothesis is a "Tauberian condition". Wiener's Tauberian theorems have the following structure:
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