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Variations and variational measures in integration theory and some applications. (English) Zbl 0959.26006

From the text: “In this survey, we consider the notions of variation and of variational measure with respect to a derivation basis and explore their feasibility by applying them to the development of an integration theory wide enough to cover many classical problems of analysis.
A concept fundamental for the whole theory is that of variational equivalence. A similar notion called “differential equivalence” and a general idea of exploiting it in defining an integral is due to Kolmogorov…Later the idea reappeared in the Kurzweil-Henstock theory of the generalized Riemann integral and the related variational integral. This integral has a simpler definition than the Lebesgue integral and yet covers a wider field, being equivalent to the Denjoy-Perron integral in the one-dimensional case. Several original papers written in the late fifties gave rise to a general theory of nonabsolutely convergent integrals.
Here our concern is to present the basic language and methods of the Henstock theory with a view to unify a variety of approaches to some problems of analysis which require integration processes more powerful than Lebesgue integration. In particular, we examine an application of generalized integrals to the problem of recovering the coefficients of an orthogonal series from its sum by generalized Fourier formulas and some applications to differential equations. We also mention the relation of the Henstock theory to the traditional theory of nonabsolute integrals…
The survey ends with a quite complete bibliography of the papers published on this subject during the last 15 years. We also touch on some new results which have not yet appeared”.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
26A45 Functions of bounded variation, generalizations
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References:

[1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshtein,Multiplicative Systems of Functions and Harmonic Analysis on Null-Dimensional Groups [in Russian], Elm, Baku (1981).
[2] F. G. Arutuhyan, ”Recovering of coefficients of series with respect to Haar and Walsh systems convergent to Denjoy integrable functions,”Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 2, 325–344 (1966).
[3] E. S. Bajgogin. ”On a dyadic Perron integral,”Vestn. Mosk. Univ. Ser. 1. Mat, Mekh.,48, No. 4, 25–28 (1993).
[4] I. A. Vinogradova and V. A. Skvortsov, ”Generalized integrals and Fourier series,” In:Itogi Nauki. Mat. Analiz (1970), All-Union Institute for Scientific and Technical Information, Moscow (1971), pp. 65–107. · Zbl 0295.42014
[5] B. S. Kashin and A. A. Saakyan,Orthogonal Series [in Russian], Nauka, Moscow (1984).
[6] V. A. Skvortsov, ”Interrelation between the general Denjoy integral and totalization (T 2s )0,”Mat. Sb.,52, No. 1, 551–578 (1960). · Zbl 0094.03504
[7] V. A. Skvortsov, ”Some properties ofCP-integrals,”Mat. Sb.,60, No. 3, 304–324 (1963); English transl. inTrans. Amer. Math. Soc.,54, 231–254 (1966). · Zbl 0145.05904
[8] V. A. Skvortsov, ”The mutual relationship between theAP-integral of Taylor and theP 2-integral of James,”Mat. Sb.,70, 380–393 (1966). · Zbl 0171.01802
[9] V. A. Skvortsov, ”On definitions ofP 2-andSCP-integrals,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh.,21, No. 6, 12–19 (1966). · Zbl 0158.30401
[10] V. A. Skvortsov, ”Calculation of the coefficients of an everywhere convergent Haar series,”Mat. Sb.,75, No. 3, 349–360 (1968); English transl. inMath. USSR Sb., 4, 317–327 (1968). · Zbl 0183.33401
[11] V. A. Skvortsov, ”Haar series with convergent subsequences of partial sums,”Dokl. Akad. Nauk SSSR,183, No. 4, 784–786 (1968); English transl. inSov. Math. Dokl.,9, 1464–1471 (1968). · Zbl 0183.33402
[12] V. A. Skvortsov, ”A generalization of the Perron integral,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh.,27, No. 4, 48–51 (1969). · Zbl 0183.31903
[13] V. A. Skvortsov, ”The Marcinkiewicz-Zygmund integral and its relation to the BurkillSCP-integral,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh.,27, No. 5, 78–82 (1972). · Zbl 0261.26007
[14] V. A. Skvortsov, ”Uniqueness theorems for Walsh series that are summable by the (C, 1) method,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh.,31, No. 5, 73–80 (1976). · Zbl 0343.42010
[15] V. A. Skvortsov, ”Constructive version of the definition of theHD-integral,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh.,37, No. 6, 41–45 (1982).
[16] V. A. Skvortsov, ”On the Marcinkiewicz theorem for the dyadic Perron integral,”Mat. Zametki,59, No. 2, 267–277 (1996). · Zbl 0878.26004
[17] V. A. Skvortsov, ”Approximate symmetric variation and the Luzin conditionN,”Izv. Ross. Akad. Nauk. Ser. Mat. (to appear).
[18] V. A. Skvortsov, ”Variational measure and a sufficient condition for the differentiability of additive interval functions,”Vestn. Mosk. Univ. Ser. 1. Mat., Mekh. (to appear). · Zbl 0913.28002
[19] V. A. Sklyarenko, ”Some properties ofP 2-primitives,”Mat. Zametki,12, 693–700 (1972); English. transl. inMath. Notes,12, 856–860 (1973). · Zbl 0249.26011
[20] V. A. Sklyarenko, ”On Denjoy integrable sums of everywhere convergent series,”Dokl. Akad. Nauk SSSR,210, No 3, 533–536 (1973); English transl. inSov. Math. Dokl.,14, 771–775 (1973). · Zbl 0283.42005
[21] Sh. T. Tetunashvili, ”On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series,”Mat. Sb.,182, No. 8 (1992); English transl. inMath. USSR Sb.,73, No. 2, 517–534 (1992). · Zbl 0772.42021
[22] G. Tolstov, ”Sur l’integrale de Perron,”Mat. Sb.,5, 647–660 (1939). · JFM 65.0200.01
[23] V. V. Filippov, ”On ordinary differential equations with singularities on the right-hand side,”Mat. Zametki,38, No. 6, 832–851 (1985). · Zbl 0626.34014
[24] V. V. Filippov, ”On existence and on properties of solutions of differential equations and differential inclusions,”Diff. Uravnenia,22, No. 6, 967–978 (1986).
[25] J. M. Ash, C. Freiling, and D. Rinne, ”Uniqueness of rectangularly convergent trigonometric series,”Ann. Math.,137, 145–166 (1993). · Zbl 0780.42015 · doi:10.2307/2946621
[26] B. Bongiorno, L. Di Piazza, and V. A. Skvortsov, ”The essential variation of a function and some convergence theorems,”Analysis Math.,22 (1996). · Zbl 0864.26006
[27] B. Bongiorno, L. Di Piazza, and V. A. Skvortsov, ”A new full descriptive characterization of the Denjoy-Perron integral,”Real Analysis Exchange,21, No. 2 (1995–96). · Zbl 0879.26026
[28] B. Bongiorno, L. Di Piazza, and V. A. Skvortsov, ”On continuous major and minor functions for then-dimensional Perron integral,”Real Analysis Exchange (to appear). · Zbl 0879.26044
[29] B. Bongiorno, W. F. Pfeffer, and B. S. Thomson, ”A full descriptive definition of the gage integral,” (to appear). · Zbl 0885.26008
[30] B. Bongiorno and V. A. Skvortsov, ”Multipliers for some generalized Riemann integrals in the real line,”Real Analysis Exchange,20, No. 1, 212–218 (1994–95). · Zbl 0828.26007
[31] Z. Buczolich, ”A general Riemann complete integral in the plane,”Acta Math. Hung.,57, No. 3-4, 315–323 (1991). · Zbl 0749.26005 · doi:10.1007/BF01903683
[32] Z. Buczolich, ”Theg-integral is not rotation invariant,”Real Analysis Exchange,18, No. 2, 437–447 (1992–93).
[33] Z. Buczolich and W. F. Pfeffer, ”Variation of additive functions” (to appear). · Zbl 0903.26004
[34] P. S. Bullen, ”Nonabsolute integrals: a survey,”Real Analysis Exchange,5, 195–259 (1979–80). · Zbl 0438.26004
[35] P. S. Bullen, ”The Burkill approximately continuous integral. II,”Math. Chron.,12, 93–98 (1983). · Zbl 0533.26006
[36] P. S. Bullen and S. N. Mukhopadhyaya, ”Integration by parts formulae for some trigonometric integrals,”Proc. London Math. Soc. (3),29, 159–173 (1974). · Zbl 0289.26011 · doi:10.1112/plms/s3-29.1.159
[37] P. Bullen and R. Vyborny, ”Some applications of a theorem of Marcinkiewicz,”Canad. Math. Bull.,43, No. 2, 165–174 (1991). · Zbl 0739.26003 · doi:10.4153/CMB-1991-027-x
[38] J. C. Burkill, ”Integrals and trigonometric series,”Proc. London Math. Soc. (3),1, 46–57 (1951). · Zbl 0042.28403 · doi:10.1112/plms/s3-1.1.46
[39] S. S. Cao, ”The Henstock integral for Banach-valued functions,”SEA Bull. Math.,16, No. 1, 35–40 (1992). · Zbl 0749.28007
[40] T. S. Chew, ”On nonlinear integrals,” In:Proc. Analysis Conf., Singapore, Vol. 150, North-Holland Math. Stud. (1986), pp. 57–62.
[41] T. S. Chew, ”On Kurzweil generalized ordinary differential equations,”J. Diff. Equations,76, 286–297 (1988). · Zbl 0666.34041 · doi:10.1016/0022-0396(88)90076-9
[42] T. S. Chew and F. Flordeliza, ”Onx’=f(t,x) and Henstock-Kurzweil integrals,”Diff. Int. Equations,4, No. 4, 861–868 (1991). · Zbl 0733.34004
[43] T. S. Chew and P. Y. Lee, ”The Henstock-Wiener integral,”J. Math. Stud.,27, No. 1, 60–65 (1994). · Zbl 0927.28009
[44] Z. T. Cong, ”On a problem of Skvortsov involving the Perron integral,”Real Analysis Exchange,17, No. 2, 748–750 (1991–92).
[45] E. A. Coddington and N. Levinson,Theory of Ordinary Differential Equations, McGraw Hill, New York (1955). · Zbl 0064.33002
[46] P. Cousin, ”Sur les fonctions den variables complexes,”Acta Math.,19, 1–61 (1895). · JFM 26.0456.02 · doi:10.1007/BF02402869
[47] G. E. Cross, ”The representation of (C, k) summable series in Fourier form,”Canad. Math. Bull.,21, 149–158 (1978). · Zbl 0386.26006 · doi:10.4153/CMB-1978-026-1
[48] G. E. Cross and B. S. Thomson, ”Symmetric integrals and trigonometric series,” In:Dissertationes Mathematicae, Vol. 319, Polska Akademia Nauk, Warszawa (1992). · Zbl 0785.26006
[49] A. Denjoy, ”Calcul des coefficients d’une séries trigonimétrique convergente quelconque dont la somme est donnée,”C. R. Acad. Sci. Paris.,172, 1218–1221 (1921). · JFM 48.0302.02
[50] A. Denjoy,Leçons sur le Calcul des Coefficients d’une Série Trigonometric, Hermann, Paris (1941–49).
[51] C. A. Edgar, ”Packing measure as a gauge variation,”Proc. Amer. Math. Soc.,122, No. 1, 167–173 (1994). · Zbl 0808.28004 · doi:10.1090/S0002-9939-1994-1197535-0
[52] H. Feizić, C. Freiling, and D. Rinne, ”Two-dimensional partitions,”Real Analysis Exchange,19, No. 2, 540–546 (1993–94).
[53] D. H. Fremlin, ”The Henstock and McShane integrals of vector-valued functions,”Ill. J. Math.,38, No. 3, 471–479 (1994). · Zbl 0797.28006
[54] D. H. Fremlin and J. Mendoza, ”On the integration of vector-valued functions,”Ill. J. Math.,38, No. 1, 127–147 (1994). · Zbl 0790.28004
[55] C. Freiling and D. Rinne, ”A symmetric density property for measurable sets,”Real Analysis Exchange,14, 203–209 (1988–89). · Zbl 0691.26005
[56] B. Golubov, A. Efimov, and V. Skvortsov,Walsh Series and Transforms, Kluwer Acad. Publ., Dordrecht (1987). · Zbl 0692.42009
[57] R. A. Gordon, ”Equivalence of the generalized Riemann and restricted Denjoy integrals,”Real Analysis Exchange,12, No. 2, 551–574 (1986–87).
[58] R. A. Gordon, ”The Denjoy extension of the Bochner, Pettis, and Dunford integrals,”Stud. Math.,92, 73–91 (1989). · Zbl 0681.28006
[59] R. A. Gordon, ”The McShane integral of Banach-valued functions,”Ill. J. Math.,34, No. 3, 557–567 (1990). · Zbl 0685.28003
[60] R. A. Gordon, ”The inversion of approximate and dyadic derivatives using an extension of the Henstock integral,”Real Analysis Exchange,16, No. 1, 154–167 (1990–91).
[61] R. A. Gordon, ”Riemann integration in Banach spaces,”Rocky Mountain J. Math.,21, No. 3, 923–949 (1991). · Zbl 0764.28008 · doi:10.1216/rmjm/1181072923
[62] R. Gordon, ”A general convergence theorem for nonabsolute integrals,”J. London Math. Soc.,44, 301–309 (1991). · Zbl 0746.26003 · doi:10.1112/jlms/s2-44.2.301
[63] M. de Guzman,Differentiation of Integrals in \(\mathbb{R}\) n , Springer-Verlag, Berlin (1975). · Zbl 0327.26010
[64] R. Henstock, ”The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals,”Proc. London Math. Soc.,10, 281–303 (1960). · Zbl 0131.29702 · doi:10.1112/plms/s3-10.1.281
[65] R. Henstock, ”Definitions of Riemann type of the variational integrals,”Proc. London Math. Soc.,11, 402–418 (1961). · Zbl 0099.27402 · doi:10.1112/plms/s3-11.1.402
[66] R. Henstock, ”Integration in product spaces, including Wiener and Feynman integration,”Proc. London Math. Soc.,27, 317–344 (1973). · Zbl 0263.28007 · doi:10.1112/plms/s3-27.2.317
[67] R. Henstock,The General Theory of Integration, Clarendon Press, Oxford (1991). · Zbl 0745.26006
[68] P. D. Humke and V. A. Skvortsov, ”Symmetric primitives and Lusin’s conditionN,”Acta Math. (to appear). · Zbl 0908.26006
[69] R. D. James, ”A generalized integral. II,”Can. J. Math.,2, 297–306 (1950). · Zbl 0037.17501 · doi:10.4153/CJM-1950-027-4
[70] J. Jarnik and J. Kurzweil, ”A nonabsolutely convergent integral which admits transformations and can be used for integration on manifolds,”Czech. Math. J.,35, 116–139 (1985). · Zbl 0614.26007
[71] J. Jarnik, J. Kurzweil, and Š. Schwabik, ”On Mawhin’s approach to multiple nonabsolutely convergent integrals,”Casopis. Pest. Mat.,108, 356–380 (1983). · Zbl 0555.26004
[72] J.-P. Kahane, ”Une théorie de Denjoy des martingales dyadiques,”L’Enseignement Math.,34, 255–268 (1988). · Zbl 0706.60043
[73] A. Kolmogorov, ”Untersuchen über den Integralbegriff,”Math. Ann.,103, 654–696 (1930). · JFM 56.0923.01 · doi:10.1007/BF01455714
[74] J. Kurzweil, ”Generalized ordinary differential equations and continuous dependence on a parameter,”Czech. Math. J.,7, 418–446 (1957). · Zbl 0090.30002
[75] J. Kurzweil and Š. Schwabik, ”Ordinary differential equations the solutions of which areACG *-functions,”Archivum Math. (Brno),26, 129–136 (1990). · Zbl 0756.34003
[76] J. Kurzweil and J. Jarnik, ”Equiintegrability and controlled convergence of Perron-type integrable functions,”Real Analysis Exchange,17, No. 1, 110–139 (1991–92).
[77] S. Leader, ”A concept of differential based on variational equivalence under generalized Riemann integration,”Real Analysis Exchange,12, No. 1, 144–175 (1986–87).
[78] S. Leader, ”Basic convergence principle for the Kurzweil-Henstock integral,”Real Analysis Exchange,18, No. 1, 95–114 (1992–93).
[79] S. Leader, ”Uniform Kurzweil-Henstock integrability,”Real Analysis Exchange,19, No. 1, 173–193 (1993–94). · Zbl 0805.26003
[80] Ch. M. Lee, ”A symmetric approximate Perron integral for the coefficient problem of convergent trigonometric series,”Real Analysis Exchange,16, No. 1, 329–339 (1990–91).
[81] P. Y. Lee,Lanzhou Lectures on Integration Theory, World Scientific, Singapore (1989). · Zbl 0699.26004
[82] J. Marcinkiewicz and A. Zygmund, ”On the differentiability of functions and the summability of trigonometrical series,”Fund. Math.,26, 1–43 (1936). · JFM 62.0287.01
[83] J. Mawhin, ”Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields,”Czech. Math. J.,31, No. 4, 614–632 (1981). · Zbl 0562.26004
[84] J. Mawhin, ”Classical problems in analysis and new integrals,”Real Analysis Exchange,12, No. 1, 69–84 (1986–87).
[85] E. McShane, ”A Riemann-type integral that includes Lebesgue, Stieltjes, Bochner, and stochastic integrals,” In:Memoirs Amer. Math. Soc., Vol. 88, Providence (1969). · Zbl 0188.35702
[86] E. McShane,Unified Integration, Academic Press, New York (1983). · Zbl 0551.28001
[87] J. Mendoza, ”On Lebesgue integrability of McShane integrable functions,”Real Analysis Exchange,18, No. 2, 456–458 (1992–93).
[88] S. Meinershagen, ”The symmetric derivation basis measure and the packing measure,”Proc. Amer. Math. Soc.,103, No. 3, 813–814 (1988). · Zbl 0657.28005 · doi:10.1090/S0002-9939-1988-0947664-6
[89] J. Mortensen, ”Advances in geometric integration,”Real Analysis Exchange,19, No. 2, 358–393 (1993–94).
[90] P. Muldowney, ”A general theory of integration in function spaces,” In:Pitman Research Notes in Math. Series, Longman Scientific and Technical, Essex (1987). · Zbl 0623.28008
[91] S. Nakanishi, ”The Henstock integral for functions with values in nuclear spaces,”Math. Jap.,39, No. 2, 309–335 (1994). · Zbl 0827.28008
[92] M. P. Navarro and V. A. Skvortsov, ”OnN-dimensional Perron integrals,”South East Asia Math. Bull.,20, No. 2, 111–116 (1996). · Zbl 0853.26008
[93] ”New integrals” [P. S. Bullen et al., eds.], In:Lect. Notes Math., Vol. 1419, Springer-Verlag (1990). · Zbl 0727.05006
[94] A. Novikov and W. Pfeffer, ”An invariant Riemann integral defined by figures,”Proc. Amer. Math. Soc.,120, 833–849 (1994). · Zbl 0808.26006 · doi:10.1090/S0002-9939-1994-1182703-4
[95] K. M. Ostaszewski, ”Henstock integration in the plane,” In:Memoirs Amer. Math. Soc., Vol. 359, Providence (1986). · Zbl 0596.26005
[96] A. Pacquement, ”Détermination d’une fonction au moyen de sa dérivée sur un réseau binaire,”C. R. Acad. Sci. Paris.,284, 365–368 (1977). · Zbl 0339.26012
[97] W. F. Pfeffer, ”The divergence theorem,”Trans. Amer. Math. Soc.,295, 665–685 (1986). · Zbl 0596.26007 · doi:10.1090/S0002-9947-1986-0833702-0
[98] W. F. Pfeffer, ”The Gauss-Green theorem,”Adv. Math.,87, 93–147 (1991). · Zbl 0732.26013 · doi:10.1016/0001-8708(91)90063-D
[99] W. F. Pfeffer, ”A descriptive definition of a variational integral and applications,”Indian Univ. Math. J.,40, No. 1, 259–270 (1991). · Zbl 0747.26010 · doi:10.1512/iumj.1991.40.40011
[100] W. F. Pfeffer,The Riemann Approach to Integration. Cambridge Univ. Press, Cambridge (1993). · Zbl 0804.26005
[101] W. F. Pfeffer, ”On variation of functions of one real variable,” (to appear in CMUC (Czech Journal)). · Zbl 0888.26006
[102] D. Preiss and B. S. Thomson, ”The approximate symmetric integral,”Can. J. Math.,41, 508–555 (1989). · Zbl 0696.26004 · doi:10.4153/CJM-1989-023-8
[103] D. Rinne, ”Rectangular and iterated convergence of multiple trigonometric series,”Real Analysis Exchange,19, No. 2, 644–650 (1993–94).
[104] S. Saks,Theory of Integral, Dover, New York (1964). · Zbl 1196.28001
[105] F. Schipp, W. R. Wade, and P. Simon,Walsh Series, Académiai Kiadó, Budapest (1990).
[106] Š. Schwabik, ”Generalized differential equations: fundamental results,”Rozpravy ČSAV,95, No. 6, 1–103 (1985).
[107] Š. Schwabik, ”The Perron integral in ordinary differential equations,”Diff. Int. Equations,6, No. 4, 863–882 (1993). · Zbl 0784.34006
[108] V. A. Skvortsov, ”Generalized integrals in the theory of trigonometric, Haar, and Walsh series,”Real Analysis Exchange,12, No. 1, 59–62 (1986–87).
[109] V. A. Skvortsov, ”A Perron-type integral in an abstract space,”Real Analysis Exchange,13, No. 1, 76–79 (1987–88).
[110] V. A. Skvortsov, ”Some properties of dyadic primitives,” In:Lect. Notes Math., Vol. 1419, Springer-Verlag (1990), pp. 167–179. · Zbl 0722.26006
[111] V. A. Skvortsov, ”Continuity of {\(\delta\)}-variation and construction of continuous major and minor functions for the Perron integral,”Real Analysis Exchange,21, No. 1, 270–277 (1995–96).
[112] V. A. Skvortsov and B. S. Thomson, ”Symmetric integrals do not have the Marcinkiewicz property,”Real Analysis Exchange,21, No. 2 (1995–96). · Zbl 0879.26030
[113] S. J. Taylor and C. Tricot, ”Packing measure and its evaluation for Brownian paths,”Trans. Amer. Math. Soc.,288, 679–699 (1985). · Zbl 0537.28003 · doi:10.1090/S0002-9947-1985-0776398-8
[114] B. S. Thomson, ”Derivation bases on the real line, I, II,”Real Analysis Exchange,8, No. 1-2, 67–207, 278–442 (1982-83). · Zbl 0525.26002
[115] B. S. Thomson, ”Derivates of interval functions.”Memoirs Amer. Math. Soc.,93, No. 452 (Providence, 1991).
[116] B. S. Thomson, ”Symmetric properties of real functions,” In:Monographs and Textbooks in Pure and Appl. Math., Vol. 183, Marcel Dekker (1994). · Zbl 0809.26001
[117] A. Zygmund,Trigonometric Series, Vol. I, Cambridge Univ. Press, London (1968). · Zbl 0157.38204
[118] A. Zygmund,Trigonometric Series, Vol. II, Cambridge Univ. Press, London (1968). · Zbl 0157.38204
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