Haar's Tauberian theorem

In mathematical analysis, Haar's Tauberian theorem[1] named after Alfréd Haar, relates the asymptotic behaviour of a continuous function to properties of its Laplace transform. It is related to the integral formulation of the Hardy–Littlewood Tauberian theorem.

Simplified version by Feller

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William Feller gives the following simplified form for this theorem:[2]

Suppose that   is a non-negative and continuous function for  , having finite Laplace transform

 

for  . Then   is well defined for any complex value of   with  . Suppose that   verifies the following conditions:

1. For   the function   (which is regular on the right half-plane  ) has continuous boundary values   as  , for   and  , furthermore for   it may be written as

 

where   has finite derivatives   and   is bounded in every finite interval;

2. The integral

 

converges uniformly with respect to   for fixed   and  ;

3.   as  , uniformly with respect to  ;

4.   tend to zero as  ;

5. The integrals

  and  

converge uniformly with respect to   for fixed  ,   and  .

Under these conditions

 

Complete version

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A more detailed version is given in.[3]

Suppose that   is a continuous function for  , having Laplace transform

 

with the following properties

1. For all values   with   the function   is regular;

2. For all  , the function  , considered as a function of the variable  , has the Fourier property ("Fourierschen Charakter besitzt") defined by Haar as for any   there is a value   such that for all  

 

whenever   or  .

3. The function   has a boundary value for   of the form

 

where   and   is an   times differentiable function of   and such that the derivative

 

is bounded on any finite interval (for the variable  )

4. The derivatives

 

for   have zero limit for   and for   has the Fourier property as defined above.

5. For sufficiently large   the following hold

 

Under the above hypotheses we have the asymptotic formula

 

References

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  1. ^ Haar, Alfred (December 1927). "Über asymptotische Entwicklungen von Funktionen". Mathematische Annalen (in German). 96 (1): 69–107. doi:10.1007/BF01209154. ISSN 0025-5831. S2CID 115615866.
  2. ^ Feller, Willy (September 1941). "On the Integral Equation of Renewal Theory". The Annals of Mathematical Statistics. 12 (3): 243–267. doi:10.1214/aoms/1177731708. ISSN 0003-4851.
  3. ^ Lipka, Stephan (1927). "Über asymptotische Entwicklungen der Mittag-Lefflerschen Funktion E_alpha(x)" (PDF). Acta Sci. Math. (Szeged). 3:4-4: 211–223.