Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-16T20:38:15.076Z Has data issue: false hasContentIssue false
  • English
  • Français

A wide Perron integral

Published online by Cambridge University Press:  17 April 2009

D. N. Sarkhel
D. N. Sarkhel
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India741235.
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 terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 2 , October 1986 , pp. 233 - 251
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Bruckner, A.M. and Ceder, J.G., “Darboux continuity”, Jahresber. Deutsch. Math.-Verein 67 (1965), 93–117.Google Scholar
[2]Bruckner, A.M., O'Malley, R.J. and Thomson, B.S., “Path derivatives: A unified view of certain generalized derivatives”, Trans. Amer. Math. Soc 283 (1984), 97–125.CrossRefGoogle Scholar
[3]Bullen, P.S., “The Pn-integral”. J. Austral. Math. Soc. 14 (1972), 219–236.CrossRefGoogle Scholar
[4]Bullen, P.S. and Lee, C.M., “On the integrals of Perron type”, Trans. Amer. Math. Soc 182 (1973), 481–501.CrossRefGoogle Scholar
[5]Bullen, P.S. and Mukhopadhyay, S.N., “Peano derivatives and general integrals”, Pacific J. Math 47 (1973), 43–58.CrossRefGoogle Scholar
[6]Burkill, J.C., “The approximately continuous Perron integral”, Math. Zeit 34 (1932), 270–278.CrossRefGoogle Scholar
[7]Burkill, J.C., “The Cesàro-Perron integral”, Proc. London Math. Soc., (2) 34 (1932), 314–322.CrossRefGoogle Scholar
[8]Burkill, J.C., “The Cesàro-Perron scale of integration”, Proc. London Math. Soc, (2) 39 (1935), 541–552.CrossRefGoogle Scholar
[9]Burkill, J.C. and Haslam-Jones, U.S., “The derivates and approximate derivates of measurable functions”, Proc. London Math. Soc, (2) 32 (1931), 346–355.CrossRefGoogle Scholar
[10]Ellis, H.W., “Mean-continuous integrals”, Canad. J. Math 1 (1949), 113–124.CrossRefGoogle Scholar
[11]Ellis, H.W., “On the compatibility of the approximate Perron and Cesàro-Perron integrals”, Proc. Amer. Math. Soc 2 (1951), 396–397.Google Scholar
[12]Kubota, Y., “An integral of the Denjoy type”, Proc. Japan Acad 40 (1964), 713–717.Google Scholar
[13]Kubota, Y., “An integral of the Denjoy type. II”, Proc. Japan Acad 42 (1966), 737–742.Google Scholar
[14]Nath, R.K. and Bose, M.K., “On the proximal Cesàro-Denjoy integral”, Soochow J. Math 10 (1984), 99–110.Google Scholar
[15]O'Malley, R.J., “Selective derivates”. Acta Math. Acad. Sci. Hungar. 29 (1977), 77–97.CrossRefGoogle Scholar
[16]Saks, S., Theory of the integral (2nd rev. ed., PWN, Warsaw, 1937).Google Scholar
[17]Sargent, W.L.C., “A descriptive definition of cesàro-Perron integrals”, Proc. London Math. Soc, (2) 47 (1941), 212–247.Google Scholar
[18]Sargent, W.L.C., “On generalized derivatives and Cesàro-Denjoy integrals”, Proc. London Math. Soc, (2) 52 (1951), 365–376.Google Scholar
[19]Sargent, W.L.C., “Some properties of C λ-continuous functions”, J. London Math. Soc 26 (1951), 116–121.CrossRefGoogle Scholar
[20]Sarkhel, D.N., “A criterion for Perron integrability”, Proc. Amer. Math. Soc 71 (1978), 109–112.CrossRefGoogle Scholar
[21]Sarkhel, D.N., “Topological aspect of Cesàro-continuity”, Proc. Indian Sc. Congress, Abstract 14 (1981).Google Scholar
[22]Sarkhel, D.N. and De, A.K., “The proximally continuous integrals”, J. Austral. Math. Soc. Ser. A 31 (1981), 26–45.CrossRefGoogle Scholar
[23]Sarkhel, D.N. and Kar, A.B., “(PVB) functions and integration”, J. Austral. Math. Soc. Ser. A 36 (1984), 335–353.CrossRefGoogle Scholar
[24]Sen, H.K., “Darboux's property and its applications”, Proc. Benares Math. Soc. (N.S.) 2 (1940), 17–23.Google Scholar
[25]Sinharoy, M., “Remarks on Darboux and mean value properties of approximate derivatives”, Comment. Math. Prace Mat 23 (1983), 315–324.Google Scholar
[26]Skvorcov, V.A., “Nekotorye svoǐstva CP-integrala”, Mat. Sb. 60 (1963), 304–324; English transl., Amer. Math. Soc. Translations, (2) 54 (1966), 231–254.Google Scholar
[27]Sunouchi, G. and Utagawa, M., “The generalized Perron integrals”, Tôhoku Math. J 1 (1949), 95–99.CrossRefGoogle Scholar
[28]Tolstoff, G., “Sur l'intégrale de Perron”, Mat. Sb 5 (1939), 647–659.Google Scholar
[29]Verblunsky, S., “On a descriptive definition of Cesàro-Perron integrals”, J. London Math. Soc, (2) 3 (1971), 326–333.CrossRefGoogle Scholar