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 $L^1(ℝ)$ 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.

Introduction

A typical Tauberian theorem is the following result, for $f\in L^1(0,\mathrm{\infty })$. If:

1. $f(x)=O(1)$ as $x\to \mathrm{\infty }$
2. $\frac{1}{x}{\int }_0^{\mathrm{\infty }}{e}^{-t/x}f(t)dt\to L$ as $x\to \mathrm{\infty }$,

then

$\frac{1}{x}{\int }_0^{x}f(t)dt\to L.$

Generalizing, let $G(t)$ be a given function, and $P_G(f)$ be the proposition

$\frac{1}{x}{\int }_0^{\mathrm{\infty }}G(t/x)f(t)dt\to L.$

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 Template:Math

Let $f\in L^1(ℝ)$ be an integrable function. The span of translations $f_a(x)=f(x+a)$ is dense in $L^1(ℝ)$ if and only if the Fourier transform of $f$ has no real zeros.

Tauberian reformulation

The following statement is equivalent to the previous result,Template:Citation needed 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 $f\ast h$ tends to zero at infinity for some $h\in L^{\mathrm{\infty }}$. Then the convolution $g\ast h$ tends to zero at infinity for any $g\in L^1$.

More generally, if

$\underset{x\to \mathrm{\infty }}{lim}(f\ast h)(x)=A\int f(x)dx$

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

$\underset{x\to \mathrm{\infty }}{lim}(g\ast h)(x)=A\int g(x)dx$

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 transform

$\phi (\theta )=\sum _{n\in \mathbb{Z}}f(n)e^{-in\theta }$

has no real zeros. The following statements are equivalent version of this result:

• Suppose the Fourier transform of $f\in l^1(\mathbb{Z})$ has no real zeros, and for some bounded sequence $h$ the convolution $f\ast h$

tends to zero at infinity. Then $g\ast h$ also tends to zero at infinity for any $g\in l^1(\mathbb{Z})$.

• Let $\phi$ be a function on the unit circle with absolutely convergent Fourier series. Then $1/\phi$ has absolutely convergent Fourier series

if and only if $\phi$ has no zeros.

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

• The maximal ideals of $A(𝕋)$ are all of the form

$M_x=\{f\in A(𝕋)\mid f(x)=0\},x\in 𝕋.$

The condition in Template:Math

Let $f\in L^2(ℝ)$ be a square-integrable function. The span of translations $f_a(x)=f(x+a)$ is dense in $L^2(ℝ)$ 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 transform

$\phi (\theta )=\sum _{n\in \mathbb{Z}}f(n)e^{-in\theta }$

has zero Lebesgue measure.

References

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External links

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