Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T02:09:40.864Z Has data issue: false hasContentIssue false
  • English
  • Français

The pn-integral

Published online by Cambridge University Press:  09 April 2009

P. S. Bullen
P. S. Bullen
Affiliation:
Department of Mathematics, The University of British ColumbiaVancouver, 8 Canada
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [5] James defined an nth order Perron integral, the Pn- ntegral, and developed its properties. His proofs are often indirect, using properties of the CkP-integrals of Burkill, [3]. In this paper a simpler definition of the Pn-integral is given — the original and not completely equivalent definition, was probably chosen as James considered this integral as a special case of one defined in terms of certain symmetric derivatives, [5], when end points of the interval of definition had naturally to be avoided. We then give direct proofs of the basic results, give a characterization of Pn-primitives, and connect the integral with certain work of Denjoy, [4].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 2 , September 1972 , pp. 219 - 236
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Bosanquet, L. S., ‘A property of Cesàro-Perron integrals’, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160–165.CrossRefGoogle Scholar
[2]Bullen, P. S., ‘A criterion for n-convexity’, Pacific Journal of Mathematics, (to appear).Google Scholar
[3]Burkill, J. C., ‘The Cesàro-Perron scale of integration’, Proc. London Math. Soc. (2) 39 (1935), 541–552.Google Scholar
[4]Denjoy, A., ‘Sur l'intégration des coefficients differentiels d'ordre superieur’, Fund. Math. 25 (1935), 273–326.Google Scholar
[5]James, R. D., ‘Generalized nth primitives’, Trans. Amer. Math. Soc. 76 (1954), 149–176.Google Scholar
[6]McGregor, J. L., An integral of Perron type (M.A. Thesis, University of British Columbia, (1951)).Google Scholar
[7]Ridder, J., ‘Ueber den Perronschen Integralbegriff und seine Beziehung zu den R, L und D Integralen’, Math. Zeit. 34 (1932), 234–239.CrossRefGoogle Scholar
[8]Saks, S., Theory of the integral (Warsaw, 1937).Google Scholar
[9]Sargent, W. L. C., ‘A descriptive definition of Cesàro-Perron integrals,’ Proc. London Math. Soc. (2) 47 (1941), 212–247.Google Scholar
[10]Sargent, W. L. C., ‘On generalized derivatives and Cesàro-Denjoy integrals’, Proc. London Math. Soc. (2) 52 (1951), 365–376.Google Scholar
[11]Skvorcov, V. A., ‘Nekotorye svoîstva CP-integrala’, Matem. Sb. 60 (102) (1963), 305–324.Google Scholar
[12]Zahorski, Z., ‘Ueber die Menge der Punkte in welchen die Ableitung unendlich ist’, Tohoku Math. J. 48 (1941), 321–330.Google Scholar