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Further results on an integral representation of functions of generalised variation

Published online by Cambridge University Press:  17 April 2009

A.M. Russell
A.M. Russell
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria.
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Abstract

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In this paper we present further properties of the kth variation of a function, and obtain an integral representation for a function having bounded kth variation and an absolutely continuous (k-l)th derivative. The absolute continuity-requirement replaces a previous stronger condition that required the kth derivative of a function to be continuous except on a set of Lebesgue measure zero.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 3 , June 1978 , pp. 407 - 420
Copyright
Copyright © Australian Mathematical Society 1978

References

[1]Hewitt, Edwin, Stromberg, Karl, Real and abstract analysis. A modern treatment of the theory of functions of a real variable (Springer-Verlag, Berlin, Heidelberg, New York, 1965. Second printing corrected: Springer-Verlag, Berlin, Heidelberg, New York, 1969. Third printing: Graduate Texts in Mathematics, 25. Springer-Verlag, New York, Heidelberg, Berlin, 1975).Google Scholar
[2]Russell, A.M., “Functions of bounded kth variation”, Proc. London Math. Soc. (3) 26 (1973), 547–563.CrossRefGoogle Scholar
[3]Russell, A.M., “An integral representation for a generalised variation of a function”, Bull. Austral. Math. Soc. 11 (1974), 225–229.CrossRefGoogle Scholar
[4]Russell, A.M., “Stieltjes-type integrals”, J. Austral. Math. Soc. Ser. A 20 (1975),431–448.CrossRefGoogle Scholar