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 or
can be approximated by linear combinations of translations of a given function.
Informally, if the Fourier transform of a function vanishes on a certain set, the Fourier transform of any linear combination of translations of also vanishes on. Therefore, the linear combinations of translations of cannot approximate a function whose Fourier transform does not vanish
on.
Wiener's theorems make this precise, stating that linear combinations of translations of are dense if and only if the zero set of the Fourier
transform of is empty or of Lebesgue measure zero.
Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the group ring
of the group of real numbers is the dual group of. A similar result is true when
is replaced by any locally compact abelian group.

Introduction

A typical Tauberian theorem is the following result, for. If:

then
Generalizing, let be a given function, and be the proposition
Note that one of the hypotheses and the conclusion of the Tauberian theorem has the form, respectively, with and
The second hypothesis is a "Tauberian condition".
Wiener's Tauberian theorems have the following structure:
Here is a "Tauberian" condition on, and is a special condition on the kernel. The power of the theorem is that holds, not for a particular kernel, but for all reasonable kernels.
The Wiener condition is roughly a condition on the zeros the Fourier transform of. For instance, for functions of class, 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 be an integrable function. The span of translations
is dense in if and only if the Fourier transform of 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 has no real zeros, and suppose the convolution
tends to zero at infinity for some. Then the convolution tends to zero at infinity for any
More generally, if
for some the Fourier transform of which has no real zeros, then also
for any.

Discrete version

Wiener's theorem has a counterpart in
has no real zeros. The following statements are equivalent version of this result:

Suppose the Fourier transform of has no real zeros, and for some bounded sequence the convolution

tends to zero at infinity. Then also tends to zero at infinity for any.

Let be a function on the unit circle with absolutely convergent Fourier series. Then has absolutely convergent Fourier series

if and only if has no zeros.
showed that this is equivalent to the following property of the Wiener algebra,
which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result:

The maximal ideals of are all of the form

The condition in

Let be a square-integrable function. The span of translations is dense in
if and only if the real zeros of the Fourier transform of form a set of zero Lebesgue measure.
The parallel statement in is as follows: the span of translations of a sequence is dense if and only if the zero set of the Fourier transform
has zero Lebesgue measure.