Geodesic
The Integration of Exact Peano Derivatives
Canadian mathematical bulletin, Tome 29 (1986) no. 3, pp. 334-340

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1986-051-8
Mots-clés : 26A24, 26A39 
Cross, G. E. The Integration of Exact Peano Derivatives. Canadian mathematical bulletin, Tome 29 (1986) no. 3, pp. 334-340. doi: 10.4153/CMB-1986-051-8
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[1] 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] 2. Burkill, J. C., The Cesàro —Perron scale of integration, Proc. London Math. Soc. (2) 39 (1935), pp. 541–552. Google Scholar

[3] 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] 4. Henstock, R., Theory of Integration, (Buttcrworth), London, 1963. Google Scholar

[5] 5. Henstock, R.. A Riemann-type integral of Lehesgue power. Can. J. Math. 20 (1968), pp. 79–87. Google Scholar

[6] 6. McLeod, R. M., The generalized Riemann integral. The Cams Mathematical Monographs (20), 1980. Google Scholar

[7] 7. Oliver, H. W., The Exact Peano Derivative, Trans. Amer. Math. Soc. 76 (1954), pp. 444–456. Google Scholar

[8] 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

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