The abscissa of absolute summability of Laplace integrals
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- by Godfrey L. Isaacs PDF
- Proc. Amer. Math. Soc. 32 (1972), 142-146 Request permission
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) \[ R(k’,w) = {(\Gamma (k’ + 1))^{ - 1}}\int _w^\infty {{{(u - w)}^{k’}}\;dA(u)\quad (k’ \geqq 0).} \] It is shown that if k is positive and fractional and if ${e^{ - ws’}}R(k,w)$ is summable $|C,0|$ for some $s’$ whose real part is negative, then $C(w)$ is summable $|C,k + \varepsilon |$ 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 $|C,k|$ summability of $\smallint _0^\infty {e^{ - us}}dA(u)$ is negative then it equals \[ \lim \sup \limits _{w \to \infty } {w^{ - 1}}\log \int _w^\infty {|dR(k,u)|} \] for all $k \geqq 0$ except one fractional value.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 32 (1972), 142-146
- MSC: Primary 44.10
- DOI: https://doi.org/10.1090/S0002-9939-1972-0290040-0
- MathSciNet review: 0290040