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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
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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
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[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