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 $L^1$ 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 $$L^1(\mathbb{R})$$ 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 $$f\in L^1(0,\infty)$$. If: then
 * 1) $$f(x)=O(1)$$ as $$x\to\infty$$
 * 2) $$\frac1x\int_0^\infty e^{-t/x}f(t)\,dt \to L$$ as $$x\to\infty$$,
 * $$\frac1x\int_0^xf(t)\,dt \to L.$$

Generalizing, let $$G(t)$$ be a given function, and $$P_G(f)$$ be the proposition
 * $$\frac1x\int_0^\infty G(t/x)f(t)\,dt \to L.$$

Note that one of the hypotheses and the conclusion of the tauberian theorem has the form $$P_G(f)$$, respectively, with $$G(t)=e^{-t}$$ and $$G(t)=1_{[0,1]}(t).$$ The second hypothesis is a "tauberian condition".

Wiener's tauberian theorems have the following structure:
 * If $$G_1$$ is a given function such that $$W(G_1)$$, $$P_{G_1}(f)$$, and $$R(f)$$, then $$P_{G_2}(f)$$ holds for all "reasonable" $$G_2$$.

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 $L^{1}$
Let $$f\in L^1(\mathbb{R})$$ be an integrable function. The span of translations $$f_a(x) = f(x+a)$$ is dense in $$L^1(\mathbb{R})$$ 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 $$f*h$$ tends to zero at infinity for some $$h\in L^\infty$$. Then the convolution $$g*h$$ tends to zero at infinity for any $$g\in L^1$$.

More generally, if


 * $$\lim_{x \to \infty} (f*h)(x) = A \int f(x) \,dx$$

for some $$f\in L^1$$ the Fourier transform of which has no real zeros, then also


 * $$\lim_{x \to \infty} (g*h)(x) = A \int g(x) \,dx$$

for any $$g\in L^1$$.

Discrete version
Wiener's theorem has a counterpart in $$l^1(\mathbb{Z})$$: the span of the translations of $$f\in l^1(\mathbb{Z})$$ is dense if and only if the Fourier series


 * $$\varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta} \,$$

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 $$g\in l^1(\mathbb{Z})$$. if and only if $$\varphi$$ has no zeros.
 * Suppose the Fourier series of $$f\in l^1(\mathbb{Z})$$ has no real zeros, and for some bounded sequence $$h$$ the convolution $$f*h$$
 * Let $$\varphi$$ be a function on the unit circle with absolutely convergent Fourier series. Then $$1/\varphi$$ has absolutely convergent Fourier series

showed that this is equivalent to the following property of the Wiener algebra $$A(\mathbb{T})$$, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:


 * The maximal ideals of $$A(\mathbb{T})$$ are all of the form


 * $$M_x = \left\{ f \in A(\mathbb{T}) \mid f(x) = 0 \right\}, \quad x \in \mathbb{T}.$$

The condition in $L^{2}$
Let $$f\in L^2(\mathbb{R})$$ be a square-integrable function. The span of translations $$f_a(x) = f(x+a) $$ is dense in $$L^2(\mathbb{R})$$ if and only if the real zeros of the Fourier transform of $$f$$ form a set of zero Lebesgue measure.

The parallel statement in $$l^2(\mathbb{Z})$$ is as follows: the span of translations of a sequence $$f\in l^2(\mathbb{Z})$$ is dense if and only if the zero set of the Fourier series


 * $$\varphi(\theta) = \sum_{n \in \mathbb{Z}} f(n) e^{-in\theta}$$

has zero Lebesgue measure.