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Section
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Mathematics
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Title
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On the general form of a linear continuous functional in the space of regulated functions
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Author(-s)
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Derr V.Ya.a
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Affiliations
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Udmurt State Universitya
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Abstract
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The author's research continues on the theory of regulated functions (functions having finite one-sided limits at each point) and $\sigma$-continuous functions (bounded functions having no more than a countable set of discontinuity points), as well as on the theory of the *-integral. The representability of a regulated function in the form of a sum of a right-continuous function and a left-continuous function is proved ($rl$-representability of the proper function).
It is shown that the general form of a linear continuous functional in the space of regulated functions ($\sigma$-continuous functions) is the *-integral of a regulated ($\sigma$-continuous) function over a function of bounded variation.
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Keywords
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regulated functions, $\sigma$-continuous functions, $rl$-representation, *-integral, linear continuous functional
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UDC
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517.518.126, 517.911
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MSC
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26B30, 26A42, 46A20
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DOI
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10.35634/vm230403
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Received
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15 September 2023
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Language
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Russian
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Citation
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Derr V.Ya. On the general form of a linear continuous functional in the space of regulated functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 4, pp. 571–580.
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References
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- 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 (in Russian). https://doi.org/10.20537/vm190201
- Derr V.Ya. On some propertits of -integral, *Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 66–89 (in Russian). https://doi.org/10.35634/vm230105
- Levin A.Yu. On the theory of ordinary differential equations. II, Vestnik Yaroslavskogo Universiteta, 1974, issue 8, pp. 122–144 (in Russian).
- Tvrdý M. Linear bounded functionals on the space of regular regulated functions, Tatra Mountains Mathematical Publications, 1996, issue 8, pp. 203–210.
- Derr V.Ya. Teoriya funktsii deistvitel'noi peremennoi. Lektsii i uprazhneniya (Theory of functions of real argument. Lectures and exercises), Moscow: Vysshaya Shkola, 2008.
- Dunford N., Schwartz J.T. Linear operators. Part 1. General theory, New York–London: Interscience Publishers, 1958.
- Derr V.Ya. On the exact pairs of classes for the Stieltjes integral, Functional Differential Equations, 2020, vol. 27, nos. 3–4, pp. 85–94.
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Full text
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