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Integration by parts

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Abstract

Many theorems of integration theory are true for a wide range of definitions of integrals. One such theorem is that giving integration by parts, and we discuss it in this paper.

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References

  1. Amemiya, I. andAndô, T.,Measure-theoretic singular integrals (in Japanese with English summary), Bull. Res. Inst. App. El., Hokkaido University13, 33–50 (1961).

    Google Scholar 

  2. Bliss, G. A.,Integrals of Lebesgue, Bull. Amer. Math. Soc.24, 1–47 (1917).

    Google Scholar 

  3. Burkill, J. C.,The Approximately Continuous Perron Integral, Math. Z.34, 270–278 (1932).

    Google Scholar 

  4. Burkill, J. C.,The Cesàro-Perron Integral, Proc. London Math. Soc. (2)34, 314–322 (1932).

    Google Scholar 

  5. Burkill, J. C.,The Cesàro-Perron Scale of Integration, Proc. London Math. Soc. (2)39, 541–552 (1935).

    Google Scholar 

  6. Burkill, J. C.,Fractional Orders of Integrability, J. London Math. Soc.11, 220–226 (1936).

    Google Scholar 

  7. Burkill, J. C.,Differential Properties of the Young-Stieltjes Integrals, J. London Math. Soc.23, 22–28 (1948).

    Google Scholar 

  8. Davies, R. O. andSchuss, Z.,A Proof that Henstock's Integral Includes Lebesgue's, J. London Math. Soc. (2)2, 561–562 (1970).

    Google Scholar 

  9. Du Bois-Reymond,Über die allgemeinen Eigenschaften der Klasse von Doppelintegralen, zu welcher das Fouriersche Doppelintegral gehört, J. Reine Angew. Math.69, 65–108 (1868).

    Google Scholar 

  10. Du Bois-Reymond, Bayer Akad. Wiss. Math. — Natur. Kl. Abh.12, 129 (1875).

    Google Scholar 

  11. Ellis, H. W.,Mean-continuous Integrals, Canad. J. Math.1, 113–124 (1949).

    Google Scholar 

  12. Foglio, S.,The N-variational Integral and the Schwartz Distributions, Proc. London Math. Soc. (3)18, 337–348 (1968).

    Google Scholar 

  13. Gordon, L. andLasher, S.,An Elementary Proof of Integration by Parts for the Perron Integral, Proc. Amer. Math. Soc.18, 394–398 (1967).

    Google Scholar 

  14. Hardy, G. G.,Notes on Some Points in the Integral Calculus (I) on the Formula for Integration by Parts, Messenger of Math.30, 185–187 (1901).

    Google Scholar 

  15. Hardy, G. H., quoted in Young [54].

  16. Hardy, G. H.,Notes on Some Points in the Integral Calculus (L) On the Integral of Stieltjes and the Formula for Integration by Parts, Messenger of Math.48, 90–100 (1918).

    Google Scholar 

  17. Hellinger, E.,Die Orthogonalinvarianten quadratischer Formen von unendlich vielen Variablen, (Diss., Göttingen, 1907).

  18. Henstock, R.,The Efficiency of Convergence Factors for Functions of a Continuous Real Variable, J. London Math. Soc.30, 273–286 (1955).

    Google Scholar 

  19. Henstock, R.,The Use of Convergence Factors in Ward Integration, Proc. London Math. Soc. (3)10, 107–121 (1960).

    Google Scholar 

  20. Henstock, R.,The Equivalence of Generalized Forms of the Ward, Variational, Denjoy-Stieltjes and Perron-Stieltjes Integrals, Proc. London Math. Soc. (3)10, 281–303 (1960).

    Google Scholar 

  21. Henstock, R.,n-variation and n-variational Integrals of Set Functions, Proc. London Math. Soc. (3)11, 109–133 (1961).

    Google Scholar 

  22. Henstock, R.,Definit i ons of Riemann Type of the Variational Integrals, Proc. London Math. Soc. (3)11, 402–418 (1961).

    Google Scholar 

  23. Henstock, R.,Theory of Integration (Butterworths, London, 1963).

    Google Scholar 

  24. Henstock, R.,A Riemann-type Integral of Lebesgue Power, Canad. J. Math.20, 79–87 (1968).

    Google Scholar 

  25. Henstock, R.,Linear Analysis (Butterworths, London, 1968).

    Google Scholar 

  26. Henstock, R.,Generalized Integrals of Vector-valued Functions, Proc. London Math. Soc. (3)19, 509–536 (1969).

    Google Scholar 

  27. Henstock, R.,Integration in Product Spaces, Including Wiener and Feynman Integration, (in manuscript).

  28. Hildebrandt, T. H.,Definitions of Stieltjes Integrals of the Riemann Type, Amer. Math. Monthly45, 265–278 (1938).

    Google Scholar 

  29. Hildebrandt, T. H.,On Systems of Linear Differentio-Stieltjes-Integral Equations, Illinois J. Math.3, 352–373 (1959).

    Google Scholar 

  30. Hobson, E. W.,On the Second Mean Value Theorem of the Integral Calculus, Proc. London Math. Soc. (2)7, 14–23 (1909).

    Google Scholar 

  31. Hobson, E. W.,The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. 1, 3rd. edition (Cambridge, 1927).

  32. Jeffery, R. L.,Perron Integrals, Bull. Amer. Math. Soc.48, 714–717 (1942).

    Google Scholar 

  33. Jeffery, R. L.,Non-absolutely Convergent Integrals, Proc. 2nd. Canad. Math. Congress 1949, 93–145 (Univ. Toronto Press, 1951).

  34. Jeffery, R. L. andMiller, D. S.,Convergence Factors for Generalized Integrals, Duke Math. J.12, 127–142 (1945).

    Google Scholar 

  35. Kolmogoroff, A.,Grundbegriffe der Wahrscheinlichkeitsrechnung, (Berlin, 1933).

  36. Krause, M.,Mittelwertsatze in Gebiete der Doppelsummen und Doppelintegrale, Leipziger Bericht, math. Phys. Klasse, 239–263 (1903).

  37. Lebesgue, H.,Sur l'integration des fonctions discontinues, Ann. Sci. École (3)27, 361–450 (1910).

    Google Scholar 

  38. Macnerney, J. S.,An Integration-by-Parts Formula, Bull. Amer. Math. Soc.69, 803–805 (1963).

    Google Scholar 

  39. McShane, E. J.,On Perron Integration, Bull. Amer. Math. Soc.48, 718–726 (1942).

    Google Scholar 

  40. McShane, E. J.,A Riemann-type Integral that Includes Lebesgue-Stieltjes, Bochner, and Stochastic Integrals, (Amer. Math. Soc. Memoirs, No. 88, 1969).

  41. Okano, H.,Sur les integrales (E.R.) et ses applications, Osaka J. Math.11, 187–212 (1959).

    Google Scholar 

  42. Okano, H.,Sur une généralisation de l'intégrale (E.R.) et un théorème général de l'intégration par parties, J. Math. Soc. Japan14, 430–442 (1962).

    Google Scholar 

  43. Perron, O.,Die Lehre von den Kettenbruchen, quoted in Pollard [44].

  44. Pollard, S.,The Stieltjes Integral and its Generalisations, Quart. J. Math. Oxford Ser.49, 87–94 (1923).

    Google Scholar 

  45. Saks, S.,Theory of the Integral, 2nd. English edition (Warsaw, 1937).

  46. Sierpinski, W.,Sur un problème concernant les ensembles mesurables superficiellement, Fund. Math.1, 112–115 (1920).

    Google Scholar 

  47. Thomae, J.,Die partielle Integration, Z.f. Math.20, 475–478 (1875).

    Google Scholar 

  48. Titchmarsh, E. C.,On Conjugate Functions, Proc. London Math. Soc. (2)29, 49–80 (1929).

    Google Scholar 

  49. Ward, A. J.,The Perron-Stieltjes Integral, Math. Z.41, 578–604 (1936).

    Google Scholar 

  50. Wright, F. M. andBaker, J. D.,On Integration-by-Parts for Weighted Integrals, Proc. Amer. Math. Soc.22, 42–52 (1969).

    Google Scholar 

  51. Young, W. H.,On the Conditions that a Trigonometrical Series Should have the Fourier Form, Proc. London Math. Soc. (2)9, 421–433 (1911).

    Google Scholar 

  52. Young, W. H.,On Integration with Respect to a Function of Bounded Variation, Proc. London Math. Soc. (2)13, 109–150 (1914).

    Google Scholar 

  53. Young, W. H.,On Non-absolutely Convergent, not Necessarily Continuous, Integrals, Proc. London Math. Soc. (2)16, 175–218 (1918).

    Google Scholar 

  54. Young, W. H.,On Multiple Integration by Parts and the Second Theorem of the Mean, Proc. London Math. Soc. (2)16, 273–293 (1918).

    Google Scholar 

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Henstock, R. Integration by parts. Aeq. Math. 9, 1–18 (1973). https://doi.org/10.1007/BF01838184

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  • DOI: https://doi.org/10.1007/BF01838184

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