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Archive of Issues

Russia Izhevsk
Year
2019
Volume
29
Issue
2
Pages
135-152
 Section Mathematics Title On the extension of a Rieman-Stieltjes integral Author(-s) Derr V.Ya.a Affiliations Udmurt State Universitya Abstract In this paper, the properties of the regular functions and the so-called $\sigma$-continuous functions (i.e., the bounded functions for which the set of discontinuity points is at most countable) are studied. It is shown that the $\sigma$-continuous functions are Riemann-Stieltjes integrable with respect to continuous functions of bounded variation. Helly's limit theorem for such functions is also proved. Moreover, Riemann-Stieltjes integration of $\sigma$-continuous functions with respect to arbitrary functions of bounded variation is considered. To this end, a $(*)$-integral is introduced. This integral consists of two terms: (i) the classical Riemann-Stieltjes integral with respect to the continuous part of a function of bounded variation, and (ii) the sum of the products of an integrand by the jumps of an integrator. In other words, the $(*)$-integral makes it possible to consider a Riemann-Stieltjes integral with a discontinuous function as an integrand or an integrator. The properties of the $(*)$-integral are studied. In particular, a formula for integration by parts, an inversion of the order of the integration theorem, and all limit theorems necessary in applications, including a limit theorem of Helly's type, are proved. Keywords functions of bounded variation, regulated functions, $\sigma$-continuous functions, Rieman-Stieltjes integral, $(*)$-integral UDC 517.518.126 MSC 26B30, 26A42 DOI 10.20537/vm190201 Received 18 March 2019 Language Russian Citation Derr V.Ya. On the extension of a Rieman-Stieltjes integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 135-152. References Kurzweil J. Linear differential equations with distributions as coefficients, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 1959, vol. 7, no. 9, pp. 557-560. Levin A.Yu. Questions on the theory of ordinary linear differential equations. II, Vestn. Yaroslav. Univ., 1974, issue 8, pp. 122-144 (in Russian). Derr V.Ya. Ordinary linear differential equations with generalized functions in coefficients: survey, Funktsional'no-differentsial'nye uravneniya: teoriya i prilozheniya (Functional differential equations: theory and applications), Perm: Perm National Research Polytechnic University, 2018, pp. 60-86 (in Russian). Atkinson F.V. Discrete and continuous boundary problems, New York: Academic Press, 1964. Translated under the title Diskretnye i nepreryvnye granichnye zadachi, Moscow: Mir, 1968. Schwabik S., Tvrdý M., Vejvoda O. Differential and integral equations. Boundary value problems and adjoints, Praha: Academia, 1979. Derr V.Ya. A generalization of Riemann-Stieltjes integral, Functional Differential Equations, 2002, vol. 9, no. 3-4, pp. 325-341. Derr V.Ya., Kinzebulatov D.M. Alpha-integral of Stieltjes type, Vestn. Udmurtsk. Univ. Mat., 2006, issue 1, pp. 41-62 (in Russian). http://mi.mathnet.ru/eng/vuu245 Rodionov V.I. The adjoint Riemann-Stieltjes integral, Russian Mathematics, 2007, vol. 51, issue 2, pp. 75-79. https://doi.org/10.3103/S1066369X07020107 Dieudonné J. Foundations of modern analysis, New York-London: Academic Press, 1960. Translated under the title Osnovy sovremennogo analiza, Moscow: Mir, 1964. Schwartz L. Analyse Mathématique. Vol. I, Paris: Hermann, 1967. Translated under the title Analiz. Tom I, Moscow: Mir, 1972. Tolstonogov A.A. Properties of the space of proper functions, Mathematical Notes of the Academy of Sciences of the USSR, 1984, vol. 35, issue 6, pp. 422-427. https://doi.org/10.1007/BF01139944 Derr V.Ya. Teoriya funktsii deistvitel'noi peremennoi. Lektsii i uprazhneniya (Theory of functions of real argument. Lectures and exercises), Moscow: Vysshaya Shkola, 2008. Derr V.Ya. Funktsional'nyi analiz. Lektsii i uprazhneniya (Functional analysis. Lectures and exercises), Moscow: Knorus, 2013. Dunford N., Schwartz J.T. Linear operators. Part I: General theory, New York-London: Interscience Publishers, 1958. Translated under the title Lineinye operatory. Tom 1. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962. Whittaker E.T., Watson G.N. A course of modern analysis, Cambridge, 1927. Translated under the title Kurs sovremennogo analisa, Moscow: Fizmatgiz, 1963. Full text