On some properties of *-integral

V. Ya. Derr

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 1, pp. 66-89

Résumé

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.

Détail

Keywords: regulated functions, $\sigma$-continuous functions, $rl$-representation, *-integral, quasi-differential equation, generalized functions, $\delta$-correctness.