Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-09T04:58:35.303Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean-Continuous Integrals

Published online by Cambridge University Press:  20 November 2018

H. W. Ellis
H. W. Ellis*
Affiliation:
The University of Toronto and Queen's University
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.

Descriptive definitions of Cesàro-Denjoy integrals (CD-integrals) equivalent to the Cesàro-Perron integrals (CP-integrals) introduced by J. C. Burkill [1, 2] have been given by Miss Sargent [6] (see §2). The CD§integrals are generalizations of the special Denjoy integral [5, p. 201]. They are somewhat complicated in that modifications of the definitions of continuity, generalized absolute continuity in the restricted sense (ACG*) [5, p. 231], and of derivatives are required for each order. In the present paper a scale of integrals is obtained which is based on the descriptive definition of the general Denjoy integral [5, p. 241]. The approximate derivative and a slightly modified definition of generalized absolute continuity (ACG) are used for all orders so that the only concept generalized for increasing orders is that of continuity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 2 , 01 April 1949 , pp. 113 - 124
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Burkill, J. C., “The Cesàro-Perron Integral,” Proc. London Math. Soc, (2), vol. 34 (1932), 314–322.Google Scholar
[2] Burkill, J. C., “The Cesàro-Perron Scale of Integration,” Proc. London Math. Soc, (2), vol. 39 (1935), 541-552.Google Scholar
[3] Jeffery, R. L. and Ellis, H. W., “Cesàro Totalization,” Trans. Royal Soc. of Canada, Third Series, Sec. Ill, vol. 36 (1942), 19-44.Google Scholar
[4] Jeffery, R. L. and Miller, D. S., “Convergence Factors for Generalized Integrals,” Duke Math. J., vol. 12 (1945), 127–142.Google Scholar
[5] Saks, S., Theory of the Integral (Warsaw, 1937).Google Scholar
[6] Sargent, W. L. C., “A Descriptive Definition of Cesàro-Perron Integrals,” Proc. London Math. Soc, (2), vol. 47 (1941), 212–247.Google Scholar
[7] Sargent, W. L. C., “On the Cesàro Derivatives of a Function,” Proc. London Math. Soc, (2), vol. 40 (1936).Google Scholar