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Integration by parts for some general integrals

Published online by Cambridge University Press:  17 April 2009

U. Das  and
A.G. Das
U. Das
Affiliation:
Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal 741235, India
A.G. Das
Affiliation:
Department of Mathematics, Faculty of Science, University of Kalyani, West Bengal 741235, India
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Abstract

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The present work is concerned with an integration by parts formula for the Pk-integral of De Sarkar and Das, and of the equivalent Pk-integral of Bullen. The process involves a simpler and updated version of that for the Zk−1-integral of Bergin. If f is Pk – (Zk−1)-integrable and G is of bounded kth variation, then fG is Pk – (Zk−1)-integrable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 1 - 15
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bergin, J.A., ‘A new characterization of Cesáro-Perron integrals using Peano derivatives’, Trans. Amer. Math. Soc. 228 (1977), 287–305.Google Scholar
[2]Bullen, P.S., ‘A criterion for n-convexity’, Pacific J. Math. 36 (1971), 81–98.Google Scholar
[3]Bullen, P.S., ‘The Pn -integral’, J. Austral. Math. Soc. 14 (1972), 219–236.Google Scholar
[4]Bullen, P.S. and Mukhopadhyay, S.N., ‘Peano derivatives and general integrals”, Pacific J. Math. 47 (1973), 43–58.Google Scholar
[5]Bullen, P.S., ‘A simple proof of integration by parts for the Perron integral’, Canad. Math. Bull 28(2) (1985), 195–199.Google Scholar
[6]Bullen, P.S., ‘A survey of integration by parts for Perron integrals’, J. Austral Math. Soc. Ser. A 40 (1986), 343–363.Google Scholar
[7]Das, U. and Das, A.G., ‘Convergence in kth variation and RSk integrals’, J. Auatral. Math. Soc. Ser A 31 (1981), 163–174.Google Scholar
[8]Das, U. and Das, A.G., ‘Approximate extenstions for Pk - and CkD-integrals’, Indian J. Math. 28(1986), 183–194.Google Scholar
[9]Das, U. and Das, A.G., ‘A new characterisation of k-fold Legesgue integral’, Comment. Math. Prace Mat. 28 (1988) (to appear).Google Scholar
[10]Das, A.G. and Lahiri, B.K., ‘On absolutely kth continuous functions’, Fund. Math. 105 (1980), 159–169.Google Scholar
[11]De Sarkar, S. and Das, A.G., ‘On fucutions of bounded kth variation’, Ind. Inst. Sc. 64(B) (1983), 299–309.Google Scholar
[12]De Sarkar, S., Das, A.G. and Lahiri, B.K., ‘Approximate Riemann* derivatives and approximate Pk-, Dk-integrals’, Indian J. Math. 27 (1985), 1–32.Google Scholar
[13]De Sarkar, S. and Das, A.G., ‘On functions of bounded essential kth variation’, Bull. Calcutta Math. Soc. 78(4) (1986), 249–258.Google Scholar
[14]De Sarkar, S. and Das, A.G., ‘Riemann derivatives and general integrals’, Bull. Austral. Math. Soc. 35 (1987), 187–211.Google Scholar
[15]Russell, A.M., ‘Functions of bounded kth variation”, Proc. London Math. Soc. (3) 26 (1973), 547–563.Google Scholar
[16]Russell, A.M., ‘Stieltjes type integrals”, J. Austral. Math. Soc. Ser. A 20 (1975), 431–448.Google Scholar
[17]Russell, A.M., ‘A Banach space of functions of generalized variation”, Bull. Austral. Math. Soc. 18 (1976), 431–438.Google Scholar
[18]Russell, A.M., ‘Further results on an integral representation of functions of generalized variation”, Bull. Austral. Math. Soc. 18 (1978), 407–420.Google Scholar
[19]Russell, A.M., ‘A commutative Banach algebra of functions of generalized variation”, Pacific J. Math. 84(2) (1979), 455–463.Google Scholar