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:

1. $f(x)=O(1)$ as $x\to \infty$
2. ${\frac {1}{x}}\int _{0}^{\infty }e^{-t/x}f(t)\,dt\to L$ as $x\to \infty$,

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

${\frac {1}{x}}\int _{0}^{\infty }G(t/x)f(t)\,dt\to L.$

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 L¹

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:

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

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.

Gelfand (1941a, 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 L²

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.

References

• Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 3–24, MR 0004726
• Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 51–66, MR 0004727
• Rudin, W. (1991), Functional analysis, International Series in Pure and Applied Mathematics, New York: McGraw-Hill, Inc., ISBN 0-07-054236-8, MR 1157815
• Wiener, N. (1932), "Tauberian Theorems", Annals of Mathematics, 33 (1): 1–100, doi:10.2307/1968102, JSTOR 1968102

External links

• Shtern, A.I. (2001) [1994], "Wiener Tauberian theorem", Encyclopedia of Mathematics, EMS Press