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 or L^2 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 cannot 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 L^1) or of Lebesgue measure zero (in the case of L^2). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L^1 group ring of the group \mathbb{R} of real numbers is the dual group of \mathbb{R}. A similar result is true when \mathbb{R} is replaced by any locally compact abelian group.

Introduction

A typical tauberian theorem is the following result, for. If: then Generalizing, let G(t) be a given function, and P_G(f) be the proposition Note that one of the hypotheses and the conclusion of the tauberian theorem has the form P_G(f), respectively, with G(t)=e^{-t} and The second hypothesis is a "tauberian condition". Wiener's tauberian theorems have the following structure: Here R(f) is a "tauberian" condition on f, and W(G_1) is a special condition on the kernel G_1. The power of the theorem is that P_{G_2}(f) holds, not for a particular kernel G_2, but for all reasonable kernels G_2. The Wiener condition is roughly a condition on the zeros the Fourier transform of G_2. For instance, for functions of class L^1, the condition is that the Fourier transform does not vanish anywhere. This condition is often easily seen to be a necessary condition for a tauberian theorem of this kind to hold. The key point is that this easy necessary condition is also sufficient.

The condition in

L{{sup|1}} Let be an integrable function. The span of translations is dense in if and only if the Fourier transform of f has no real zeros.

Tauberian reformulation

The following statement is equivalent to the previous result, and explains why Wiener's result is a Tauberian theorem: Suppose the Fourier transform of f\in L^1 has no real zeros, and suppose the convolution fh tends to zero at infinity for some. Then the convolution gh tends to zero at infinity for any g\in L^1. More generally, if for some f\in L^1 the Fourier transform of which has no real zeros, then also for any g\in L^1.

Discrete version

Wiener's theorem has a counterpart in has no real zeros. The following statements are equivalent version of this result: tends to zero at infinity. Then g*h also tends to zero at infinity for any. if and only if \varphi has no zeros. showed that this is equivalent to the following property of the Wiener algebra , which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:

The condition in

L{{sup|2}} Let be a square-integrable function. The span of translations is dense in if and only if the real zeros of the Fourier transform of f form a set of zero Lebesgue measure. The parallel statement in is as follows: the span of translations of a sequence is dense if and only if the zero set of the Fourier series has zero Lebesgue measure.