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A survey of intergration by parts for Perron integrals

Part of: Measure and integration

Published online by Cambridge University Press:  09 April 2009

P. S. Bullen
P. S. Bullen
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver B. C. V6T 1W5, Canada
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Abstract

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The history of the proof of the integration by parts formula for the Perron integral, and for the SCP-integral of Burkill, is discussed.

MSC classification

Secondary: 28-02: Research exposition (monographs, survey articles)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 3 , June 1986 , pp. 343 - 363
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Aleksandrov, A., ‘Über die Äquivalenz des Perronschen und des Denjoyschen Integralbegriffes’, Math. Z. 20 (1924), 213–222.Google Scholar
[2]Bauer, H., ‘Der Perronschen Integralbegriff und seine Beziehung zum Lebesgueschen’, Monatsch. Math. Phys. 26 (1915), 153–198.Google Scholar
[3]Bruckner, Andrew M., Differentiation of real functions (Lecture Notes in Math. 659, Springer-Verlag, New York, 1978).CrossRefGoogle Scholar
[4]Bullen, P. S. ‘Non-absolute integrals: a survey’, Real Anal. Exchange 5 (1980), 195–259.Google Scholar
[5]Bullen, P. S. and Mukhopadhyay, S. N., ‘Integration by parts formula for some trigonometric integrals’, Proc. London Math. Soc. (3) 28 (1974), 159–173.CrossRefGoogle Scholar
[6]Burkill, J. C., ‘The Cesàro-Perron scale of integration’, Proc. London Math. Soc. (2) 39 (1935), 541–552.Google Scholar
[7]Burkill, J. C., ‘Integrals and trigonometric series’, Proc. London Math. Soc. (3) 1 (1951), 46–57.CrossRefGoogle Scholar
Corrigendum, Proc. London Math. Soc. (3) 46 (1983), 190.Google Scholar
[8]Cross, G., ‘A relation between two symmetric integrals’, Proc. Amer. Math. Soc. 14 (1963), 185–190.Google Scholar
[9]Denjoy, Arnaud, ‘Une extension de l'intègrale de M. Lebesgue’, C. R. Acad. Sci. Paris 154 (1912), 859–862.Google Scholar
[10]Denjoy, Arnaud, Leçons sur le calcul de coefficients d'une série trigonométrique I–IV (Gautheir Villars, Paris, 1941–1949).Google Scholar
[11]Denjoy, Arnad, Mémorie sur la dérivation et son calcul inverse (Gautheir Villars, Paris, 1954).Google Scholar
[12]Gage, W. H. and James, R. D., ‘A generalized integral’, Trans. Roy. Soc. Canada III (3) 40 (1946), 25–35.Google Scholar
[13]Gordon, L. and Lasher, S., ‘An elementary proof of integration by parts for the Perron integral’, Proc. Amer. Math. Soc. 18 (1967), 394–398.CrossRefGoogle Scholar
[14]Hake, H., ‘Über de la Vallée Poussins Ober-und Unterfunktionen’, Math. Ann. 83 (1921), 119–142.CrossRefGoogle Scholar
[15]Henstock, R., ‘Integration by parts’, Aequationes Math. 9 (1973), 1–18.Google Scholar
[16]Hobson, E. W., The theory of functions of a real variable (Cambridge University Press, 3rd ed., 1927).Google Scholar
[17]Ionescu-Tulcea, C. T., ‘Sur l'intègration des nombres derivès’, C. R. Acad. Sci. Paris 225 (1949), 558–560.Google Scholar
[18]James, R. D., ‘A generalized integral II’, Canad. J. Math. 2 (1950), 297–306.Google Scholar
[19]James, R. D., ‘Integrals and summable trigonometric series’, Bull. Amer. Math. Soc. 61 (1955), 1–15.Google Scholar
[20]Jeffrey, R. L., ‘Perron integrals’, Bull. Amer. Math. Soc. 48 (1942), 714–1–717.Google Scholar
[21]Lebesgue, H., Leçons sur l'intégration et la recgerche des fonctions primitives (2nd ed., Paris, 1926).Google Scholar
[22]Looman, H., ‘Über die Perronsche integral definition’, Math. Ann. 93 (1935), 153–156.Google Scholar
[23]Luzin, N. N., ‘Sur les propriétés de l'integral de M. Denjoy’, C. R. Acad. Sci. Paris 155 (1912), 1475–1478.Google Scholar
[24]Maˇik, J., ‘Základy theorie integr´lu v euklidových prostorech, I–III’, Časopis Pěst. Math. 77 (1952), 1–51, 125–145, 267–230.Google Scholar
[25]McShane, Edward James, ‘On Perron integration’, Bull. Amer. Math. Soc. 48 (1942), 718–726.Google Scholar
[26]McShane, Edward James, Integration (Princeton University Press, 1944).Google Scholar
[27]Natanson, I. P., Theory of functions of a real variable (2nd ed., revised, New York, 1960).Google Scholar
[28]Perron, O., ‘Über den Integralbelgriff, Sitzber’, Heidelberg Akad. Wiss. Abt. A 16 (1914), 1–16.Google Scholar
[29]Pesin, I. N., Classical and modern integration theory (New York, 1970).Google Scholar
[30]Pfeffer, W., ‘Integration by parts for the generalized Riemann-Stieltjes integral’, J. Austral. Math. Soc. (SeriesA) 34 (1983), 229–233.Google Scholar
[31]Ridder, J., ‘Ueber Definitionen von Perron-Integralen’, I, II, Indag.Math. 9 (1947), 227–235, 280–289; MR 8, 506; 9, 19.Google Scholar
[32]Saks, S., Theory of the integral, (2nd ed., revised, New York, 1937) Zbl. 17, 300.Google Scholar
[33]Sargent, W. L. C., ‘On the integrability of a product’, J. London Math. Soc. 23 (1948), 28–34. MR 10, 108.CrossRefGoogle Scholar
[34]Sarkhel, D. N., ‘A criterion for Perron integrability’, Proc. Amer. Math. Soc. 7 (1978), 109–112. MR 58 # 17006.Google Scholar
[35]Sklyarenko, V. A., ‘Some properties of the P2-integral’, Mat. Zamerki 12 (1972), 693–700;Google Scholar
English transl., Math. Notes 12 (1974), 856–860 (1973). MR 47 # 8785.Google Scholar
[36]Sklyarenko, V. A., ‘On integration by parts for Burkill's SCP-integrals’, Math. Sb. 112 (1980), 630–646;Google Scholar
nglish transl., Math. USSR Sbornik 40 (1981), 567–583. MR 81k, 26009.Google Scholar
[37]Tolstov, G. P., ‘Sur l'intégrale de Perron’, Mat. Sb. 5 (1939), 647–659: 1. 208.Google Scholar
[38]Zygmund, A., Trigonometrical series, (Cambridge Univ. Press, 1959). MR 21 # 6498.Google Scholar