Wiener's Tauberian theorem

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 or can be approximated by linear combinations of translations of a given function.[2]

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 (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 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

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

  1.   as  
  2.   as  ,

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:[3]

If   is a given function such that  ,  , and  , then   holds for all "reasonable"  .

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 L1

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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

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The following statement is equivalent to the previous result,[citation needed] 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

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

 

has no real zeros. The following statements are equivalent version of this result:

  • Suppose the Fourier series 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.

Gelfand (1941a, 1941b) 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 L2

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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 series

 

has zero Lebesgue measure.

Notes

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  1. ^ See Wiener (1932).
  2. ^ see Rudin (1991).
  3. ^ G H Hardy, Divergent series, pp 385-377

References

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