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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An extension of the integral test


Authors: Ana Lía Durán and Ricardo Estrada
Journal: Proc. Amer. Math. Soc. 127 (1999), 1745-1751
MSC (1991): Primary 26A06, 40A10, 46F10
Published electronically: February 11, 1999
MathSciNet review: 1610952
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Abstract | References | Similar Articles | Additional Information

Abstract: We identify a class of functions for which an extension of the well known integral test holds, namely, the series $\sum _{k=1}^{\infty }\phi (k)$ is Cesàro summable if and only if the integral $\int _{1}^{\infty }\phi (x)\,dx$ is. We also give the corresponding multidimensional results.


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

  • 1. J. Boersma, A nonharmonic trigonometric series, solution of Problem 94-12, SIAM Review 37 (1995), 443-445.
  • 2. A. L. Durán and R. Estrada, The analytic continuation of moment functions and of zeta functions, preprint, San José, 1997.
  • 3. A. L. Durán, R. Estrada and R. P. Kanwal, Extensions of the Poisson summation formula, J. Math. Anal. Appl. 218 (1998), 581-606. CMP 98:08
  • 4. R. Estrada, The Cesàro behavior of distributions, Proc. Roy. Soc. London A (to appear).
  • 5. R. Estrada, J. M. Gracia-Bondía and J. C. Várilly, On summability of distributions and spectral geometry, Commun. Math. Phys. 191 (1998), 219-248. CMP 98:07
  • 6. Ricardo Estrada and Ram P. Kanwal, Asymptotic analysis, Birkhäuser Boston, Inc., Boston, MA, 1994. A distributional approach. MR 1254657 (95g:46071)
  • 7. A. Grossmann, G. Loupias, and E. M. Stein, An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), no. fasc. 2, 343–368, viii (1969) (English, with French summary). MR 0267425 (42 #2327)
  • 8. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620 (11,25a)
  • 9. Ram P. Kanwal, Generalized functions, Mathematics in Science and Engineering, vol. 171, Academic Press, Inc., Orlando, FL, 1983. Theory and technique. MR 732788 (85f:46001)
  • 10. Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966 (French). MR 0209834 (35 #730)
  • 11. E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1939.

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Additional Information

Ana Lía Durán
Affiliation: Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica
Email: analiad@emate.ucr.ac.cr

Ricardo Estrada
Affiliation: Escuela de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica
Email: restrada@cariari.ucr.ac.cr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04911-4
PII: S 0002-9939(99)04911-4
Keywords: Integral, series, summability
Received by editor(s): September 12, 1997
Published electronically: February 11, 1999
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society