Wiener's Tauberian theorem explained

In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.^[1] 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.^[2]

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,infty)} . If:f(x) = O( 1 ) as x\to inf ty (1 x\int)/(_{0}) ^{infty} e ^{-t/x} f(t)dt \to L as x\to inf ty , then(1 x\int)/(_{0}) ^{xf(t)dt} \to L.

Generalizing, let

G(t) be a given function, and P _{G(f)} be the proposition(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:^[3]

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

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, 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 anyg\in L ^{1} .

More generally, if

\lim _{x} (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} (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(Z)}

the span of the translations of

f\in l ^{1(Z)} is dense if and only if the Fourier series\varphi(\theta) = \sum _{n} 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(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(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.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 formM _{x} = \left\{ f \in A( T ) \mid f(x) = 0 \right\}, x \in T .

The condition in

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(Z)} is as follows: the span of translations of a sequence f\in l ^{2(Z)} is dense if and only if the zero set of the Fourier series\varphi(\theta) = \sum _{n} f(n) e ^{-in\theta}

has zero Lebesgue measure.

Notes and References

1. See .
2. see .
3. , pp 385-377