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A Tauberian Theorem of Exponential Type

Published online by Cambridge University Press:  20 November 2018

J. L. Geluk ,
L. de Haan  and
U. Stadtmüller
J. L. Geluk
Affiliation:
Erasmus Universiteit, Rotterdam, Netherlands
L. de Haan
Affiliation:
Universität Ulm, Ulm, West Germany
U. Stadtmüller
Affiliation:
Universität Ulm, Ulm, West Germany
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1. Introduction. We will be interested in Tauberian theorems concerning the limiting behaviour of a monotone function U and its Laplace transform

A famous theorem of Karamata concerns the case in which the function U is regularly varying (i.e., U(tx)/U(t) → xα(t → ∞) for x > 0). Here we will consider functions U that grow faster, in fact our conditions will be in terms of log U rather than on U itself. So it is convenient to write the Laplace transform in terms of q = log U. For a function q:R+ → R such that exp q is locally integrable and

we define the function by the relation

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 3 , 01 June 1986 , pp. 697 - 718
Copyright
Copyright © Canadian Mathematical Society 1986

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