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
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
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 following asymptotic formula
References
- Haar, Alfred (December 1927). "Über asymptotische Entwicklungen von Funktionen". Mathematische Annalen (in German). 96 (1): 69–107. doi:10.1007/BF01209154. ISSN 0025-5831.
- 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.
- Lipka, Stephan (1927). "Über asymptotische Entwicklungen der Mittag-Lefflerschen Funktion E_alpha(x)" (PDF). Acta Sci. Math. (Szeged). 3:4-4: 211–223.