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.see .

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 can not 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 (in the case of ) or of Lebesgue measure zero (in the case of ).

Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L¹ group ring L¹(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.

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

: $\lim_ (f*h)(x) = A \int f(x) \,dx$

for some the Fourier transform of which has no real zeros, then also

: $\lim_ (g*h)(x) = A \int g(x) \,dx$

for any .

Discrete version

Wiener's theorem has a counterpart in : the span of the translations of is dense if and only if the Fourier transform

:$\varphi(\theta) = \sum_ f(n) e^ \,$

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

::$M_x = \left\, \quad x \in \mathbb.$

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

:$\varphi(\theta) = \sum_ f(n) e^$

has zero Lebesgue measure.

Notes

References

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External links

*{{eom, id=W/w097950, title=Wiener Tauberian theorem, first=A.I., last=Shtern

Real analysis
Harmonic analysis
Tauberian theorems