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The Integration of Exact Peano Derivatives

Published online by Cambridge University Press:  20 November 2018

G. E. Cross
G. E. Cross*
Affiliation:
Department of Purk Mathematics, University of WaterlooWaterloo, Ontario N2L 3G1
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Abstract

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It is well known that the Riemann-complete integral (or equivalently the Perron integral) integrates an everywhere finite ordinary first derivative (which may be thought of as a Peano derivative of order one). It is also known that the Cesàro-Perron integral of order (n - 1) integrates an everywhere finite Peano derivative of order n. The present work concerns itself with necessary and sufficient conditions for the Riemann-complete integrability of an exact Peano derivative of order n. It is shown that when the integral exists, it can be expressed as the ‘Henstock' limit of the sum of a particular kind of interval function. All functions considered will be real valued.

Keywords

26A24, 26A39
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 3 , 01 September 1986 , pp. 334 - 340
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Bergin, J. A., A new characterization of Cesàro —Perron integrals using Peano derivatives, Trans. Amer. Math. Soc, 228 (1977), pp. 287–305.Google Scholar
2. Burkill, J. C., The Cesàro —Perron scale of integration, Proc. London Math. Soc. (2) 39 (1935), pp. 541–552.Google Scholar
3. Davics, R. O. and Schuss, Z., A proof that Henstock's integral includes Lebesgue's, J. London Math. Soc. (2) 2 (1970), pp. 561–562.Google Scholar
4. Henstock, R., Theory of Integration, (Buttcrworth), London, 1963.Google Scholar
5. Henstock, R.. A Riemann-type integral of Lehesgue power. Can. J. Math. 20 (1968), pp. 79–87.Google Scholar
6. McLeod, R. M., The generalized Riemann integral. The Cams Mathematical Monographs (20), 1980.Google Scholar
7. Oliver, H. W., The Exact Peano Derivative, Trans. Amer. Math. Soc. 76 (1954), pp. 444–456.Google Scholar
8. Yee, L. P. and Naak-In, W., A direct proof that Henstock and Denjoy integrals are equivalent. Bull. Malaysian Math. Soc. (2) 5 (1982), pp. 43–47.Google Scholar