Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1972-0290040-0
MathSciNet review: 0290040
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] L. S. Bosanquet, The summability of Laplace-Stieltjes integrals, Proc. London Math. Soc. (3) 3 (1953), 267-304. MR 15, 307. MR 0058010 (15:307d)
  • [2] -, The summability of Laplace-Stieltjes integrals. II, Proc. London Math. Soc. (3) 11 (1961), 654-690. MR 25 #5352. MR 0141957 (25:5352)
  • [3] G. L. Isaacs, An extension of a limitation theorem of M. Riesz, J. London Math. Soc. 33 (1958), 406-418. MR 21 #270. MR 0101459 (21:270)
  • [4] -, On the summability-abscissae of Laplace integrals, Proc. London Math. Soc. (3) 10 (1960), 461-479. MR 22 #9815. MR 0119048 (22:9815)
  • [5] -, The iteration formula for inverted fractional integrals, Proc. London Math. Soc. (3) 11 (1961), 213-238. MR 23 #A1979. MR 0124667 (23:A1979)
  • [6] -, A limitation theorem for absolute summability, Proc. Amer. Math. Soc. 29 (1971), 47-54. MR 0277955 (43:3688)
  • [7] A. Peyerimhoff, Über einen absoluten Mittelwertsatz und Konvexitätssatz für Rieszsche Mittel, Math. Ann. 157 (1964), 42-64. MR 30 #1337. MR 0171106 (30:1337)
  • [8] B. Prasad, Certain aspects of the researches on Dirichlet series, Twenty-Seventh Conference, Indian Mathematical Society, Ahmedabad, India, 1961.
  • [9] P. Srivastava, On the abscissa of absolute summability of a Dirichlet series, Indian J. Math. 1 (1959), 77-86. MR 22 #6977. MR 0116182 (22:6977)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 44.10

Retrieve articles in all journals with MSC: 44.10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0290040-0
Keywords: Laplace integral, summable $ \vert C,k\vert$, abscissa of summability
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society