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Haar's Tauberian theorem

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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.