Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T13:27:23.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence in kth variation and RSk integrals

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

U. Das  and
A. G. Das
U. Das
Affiliation:
Department of Mathematics, University of Kalyani, West Bengal, India
A. G. Das
Affiliation:
Department of Mathematics, University of Kalyani, West Bengal, India
Rights & Permissions [Opens in a new window]

Abstract

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 recent papers, Russell introduced the notions of functions of bounded kth variation (BVk functions) and the RSk integral. Das and Lahiri enriched Russell's works along with a convergence formula of RSk integrals depending on the convergence of integrands. In this paper a convergence theorem analogous to Arzela's dominated convergence theorem has been presented. An investigation to the convergence in kth variation has been made leading to some convergence theorems of RSk integrals depending on the convergence of integrators.

MSC classification

Secondary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type 26A45: Functions of bounded variation, generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 2 , August 1981 , pp. 163 - 174
Copyright
Copyright © Australian Mathematical Society 1982

References

Bullen, P. S. (1971), ‘A criterion for n convexity’, Pacific J. Math. 36, 81–98.CrossRefGoogle Scholar
Das, A. G. and Lahiri, B. K. (1980a), ‘On absolutely kth continuous functions’, Fund. Math. 105, 159–169.CrossRefGoogle Scholar
Das, A. G. and Lahiri, B. K. (1980b), ‘On RSk integrals’, Comment. Math. Univ. Carolinae, Sec. 1, 22, to appear.Google Scholar
Luxemburg, W. A. J. (1971), ‘Arzela's dominated convergence theorem for the Riemann integral’, Amer. Math. Monthly 78, 970–979.Google Scholar
Milne-Thomson, L. M. (1965), The calculus of finite differences (Macmillan).Google Scholar
Natanson, I. P. (1961), Theory of functions of a real variable, Volume I (Ungar, revised addition).Google Scholar
Russell, A. M. (1973), ‘Functions of bounded kth variation’, Proc. London Math. Soc. (3), 26, 547–563.CrossRefGoogle Scholar
Russell, A. M. (1975), ‘Stieltjes-type integrals’, J. Austral. Math. Soc. Ser. A 20, 431–448.CrossRefGoogle Scholar