Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T21:46:23.587Z Has data issue: false hasContentIssue false
  • English
  • Français

On ω-approximately continuous Perron-Stieltjes and Denjoy-Stieltjes integral

Published online by Cambridge University Press:  09 April 2009

D. N. Sarkhel
D. N. Sarkhel
Affiliation:
Department of Mathematics, R. K. Mission Vidyamandira, Belur Math. Howrah, West-Bengal, India.
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.

The aim of the present paper is to introduce a definition of the Perron-Stieltjes integral employing the notion of approximate derivative with respect to a nondecreasing function ω and to study some of the properties of the integral. Various authors have studied the Perron integral and Perron-Stieltjes integral in different ways, most of which can be found in the references appended in the list of the bibliography. Among them Ridder [10] uses the concept of approximate co-derivative but he assumes that the monotone function a associated with co is continuous. Finally we consider a more general type of integral, the co-approximately continuous Denjoy-Stieltjes integral, defined descriptively by the method of Saks [11].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 129 - 152
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Blumberg, H., ‘The measurable boundaries of an arbitrary function’, Acta Math. 65 (1935), 263–282.CrossRefGoogle Scholar
[2]Burkill, J. C., ‘The approximately continuous Perron Integral’, Math. Zeit. 34 (1931) 270–278.CrossRefGoogle Scholar
[3]Burkill, J. C. and Jones, U.S. Haslam, ‘The derivates and approximate derivates of measurable functions’, Proc. Lond. Math. Soc. (2) 32 (1931), 346–355.CrossRefGoogle Scholar
[4]Chakrabarty, M. C., ‘Some results on ωderivatives and BV-ω functions’, J. Aust. Math. Soc. 9 (1969), 345–360.CrossRefGoogle Scholar
[5]Chow, Shu-ER, ‘On approximate derivatives’, Bull. Amer. Math. Soc. 54 (1948), 793–802.CrossRefGoogle Scholar
[6]Jeffery, R. L., ‘Generalized integrals with respect to functions of bounded variation’, Canad. J. Math. 10 (1958), 617–626.CrossRefGoogle Scholar
[7]Jeffery, R. L., The theory of functions of a real variable (University of Toronto Press, 1962).Google Scholar
[8]Kubota, Yôto, ‘On the approximately continuous Denjoy Integrals’, Tohoku Math. Jour. 15 (1963), 253–264.Google Scholar
[9]Natanson, I. P.Theory of functions of a real variable, Vol. II. (New York, 1960), Ungar.Google Scholar
[10]Ridder, J., ‘Ueber Perron – Stieltjessche und Denjoy – Stieltjessche Integrationen’, Math. Zeit. 40 (1936), 127–160.CrossRefGoogle Scholar
[11]Saks, S., Theory of the integral, (Dover, New York, 1964).Google Scholar
[12]Sonouchi, G. and Utagawa, M., ‘The generalized Perron Integrals’, Tohoku Math. Jour. 1 (1949), 95–99.Google Scholar