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
$L^1(\mathbb)$ of the group $\mathbb$ of real numbers is the dual group of $\mathbb$. A similar result is true when
$\mathbb$ 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:
# $f(x)=O(1)$ as $x\to\infty$
# $\frac1x\int_0^\infty e^f(t)\,dt \to L$ as $x\to\infty$,
then
:$\frac1x\int_0^xf(t)\,dt \to L.$

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 $G(t)=e^$ and $G(t)=1_(t).$
The second hypothesis is a "Tauberian condition".

Wiener's Tauberian theorems have the following structure:, pp 385-377
:If $G_1$ is a given function such that $W(G_1)$, $P_(f)$, and $R(f)$, then $P_(f)$ 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 $P_(f)$ 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

Let $f\in L^1(\mathbb)$ be an integrable function. The span of translations $f_a(x) = f(x+a)$
is dense in $L^1(\mathbb)$ 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

: $\lim_ (f*h)(x) = A \int f(x) \,dx$

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

: $\lim_ (g*h)(x) = A \int g(x) \,dx$

for any $g\in L^1$.

Discrete version

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

:$\varphi(\theta) = \sum_ f(n) e^ \,$

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

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

showed that this is equivalent to the following property of the Wiener algebra $A(\mathbb)$,
which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:

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

::$M_x = \left\, \quad x \in \mathbb.$

The condition in

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

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

:$\varphi(\theta) = \sum_ f(n) e^$

has zero Lebesgue measure.

Notes

References

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External links

*{{eom, id=W/w097950, title=Wiener Tauberian theorem, first=A.I., last=Shtern

Real analysis
Harmonic analysis
Tauberian theorems