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Section
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Mathematics
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Title
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On some properties of *-integral
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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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This work continues the author's research on the theory of regulated functions and *-integral. The possibility to express a regulated function as a sum of right-continuous and left-continuous functions (called $rl$-representation) is studied. A limit theorem for the *-integral is proved. It allows approximating discontinuous integrands and integrators by sequences of absolutely continuous functions. A new result on $\delta$-correctness of the solution of an ordinary linear differential equation with generalized functions in coefficients is proved. This solution is defined via a quasi-differential equation. A formula for the total variation of an indefinite *-integral of a $\sigma$-continuous function with respect to a function of bounded variation is given. It generalizes the well-known formula for computing the total variation of an absolutely continuous function. The formula is also interesting in the case of an indefinite $RS$-integral.
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Keywords
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regulated functions, $\sigma$-continuous functions, $rl$-representation, *-integral, quasi-differential equation, generalized functions, $\delta$-correctness
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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, 34K10
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DOI
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10.35634/vm230105
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Received
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20 December 2022
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Language
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Russian
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Citation
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Derr V.Ya. On some properties of *-integral, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 66-89.
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References
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