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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The abscissa of absolute summability of Laplace integrals

Author: Godfrey L. Isaacs
Journal: Proc. Amer. Math. Soc. 32 (1972), 142-146
MSC: Primary 44.10
MathSciNet review: 0290040
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Abstract: With $ A(u)$ of bounded variation over every finite interval of the nonnegative real axis, we write $ C(w) = \smallint _0^w{e^{ - us}}dA(u)$ and (formally)

$\displaystyle R(k',w) = {(\Gamma (k' + 1))^{ - 1}}\int_w^\infty {{{(u - w)}^{k'}}\;dA(u)\quad (k' \geqq 0).} $

It is shown that if k is positive and fractional and if $ {e^{ - ws'}}R(k,w)$ is summable $ \vert C,0\vert$ for some $ s'$ whose real part is negative, then $ C(w)$ is summable $ \vert C,k + \varepsilon \vert$ for each $ \varepsilon > 0$, where s is such that its real part is greater than that of $ s'$; if k is nonnegative and integral the result holds with $ \varepsilon = 0$. Together with a 'converse' result, this may be used to show that if the abscissa of $ \vert C,k\vert$ summability of $ \smallint _0^\infty {e^{ - us}}dA(u)$ is negative then it equals

$\displaystyle \mathop {\lim \sup }\limits_{w \to \infty } {w^{ - 1}}\log \int_w^\infty {\vert dR(k,u)\vert} $

for all $ k \geqq 0$ except one fractional value.

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Keywords: Laplace integral, summable $ \vert C,k\vert$, abscissa of summability
Article copyright: © Copyright 1972 American Mathematical Society