Wiener's Tauberian theorem

In mathematical analysis, Wiener's Tauberian theorem is any of several related results proved by Norbert Wiener in 1932.See {{harvtxt|Wiener|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.see {{harvtxt|Rudin|1991}}.

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:

$f(x)=O(1)$ as $x\to\infty$
$\frac1x\int_0^\infty e^{-t/x}f(t)\,dt \to L$ 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 $G(t)=e^{-t}$ and $G(t)=1_{[0,1]}(t).$

The second hypothesis is a "Tauberian condition".

Wiener's Tauberian theorems have the following structure:{{citation|author=G H Hardy|title=Divergent series}}, pp 385-377

: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 {{math|''L''<sup>1</sup>}}

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,{{Citation needed|date=October 2018}} 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 transform

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

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

Suppose the Fourier transform of $f\in l^1(\mathbb{Z})$ 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 $g\in l^1(\mathbb{Z})$ .

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.

{{harvs|txt|last=Gelfand|year1=1941a|year2=1941b}} 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 {{math|''L''<sup>2</sup>}}

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 transform

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

has zero Lebesgue measure.

Notes

References

{{Citation | last1=Gelfand | first1=I. | author-link=Israel Gelfand|title=Normierte Ringe | year=1941a | journal=Rec. Math. (Mat. Sbornik) |series=Nouvelle Série| volume=9 | issue = 51 | pages=3–24 | mr=0004726}}
{{Citation | last1=Gelfand | first1=I. | author-link=Israel Gelfand|title=Über absolut konvergente trigonometrische Reihen und Integrale | year=1941b | journal=Rec. Math. (Mat. Sbornik) |series=Nouvelle Série| volume=9 | issue = 51 | pages=51–66 | mr=0004727}}
{{Citation|mr=1157815|last=Rudin|first=W.|author-link=Walter Rudin|title=Functional analysis|series=International Series in Pure and Applied Mathematics|publisher=McGraw-Hill, Inc.|location=New York|year=1991|isbn=0-07-054236-8|url-access=registration|url=https://archive.org/details/functionalanalys00rudi}}
{{Citation | last=Wiener|first=N.|author-link=Norbert Wiener|title=Tauberian Theorems|journal=Annals of Mathematics|volume=33|issue=1|year=1932|pages=1–100|jstor=1968102|doi=10.2307/1968102}}

External links

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