Wiener's Tauberian theorem

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. \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: :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''¹}}

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 fh tends to zero at infinity for some h\in L^\infty. Then the convolution gh 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 fh tends to zero at infinity. Then gh 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.

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