On a modification of fractional Riemann-Liouville integrodifferentiation, which is applicable to functions on \(R^ 1\) of arbitrary behavior at infinity (Q2366418)

The paper deals with a modification of Liouville's fractional integro- differentiation on the real axis related to a fixed point of the axis. Fractional integration in such a form was introduced by \textit{Y. W. Chen} [Commun. Pure Appl. Math. 14, 229-255 (1961; Zbl 0102.08903)] who has considered the corresponding fractional differentiation in the Riemann-Liouville form. Fractional differentiation of this type in the Marchaud form was investigated by \textit{A. V. Skorikov} [Mat. Zametki 17, 691-701 (1975; Zbl 0312.46040)] who has used these constructions called Chen- Marchaud derivatives to characterize the space of functions which are restrictions of the Bessel potentials of \(p\)-summable functions on the whole line onto a finite interval. In this connection see sections 18.5 and 23.2 in the book of the first author, the reviewer and \textit{O. I. Marichev} [Integrals and derivatives of fractional order and some of their applications (1987; Zbl 0617.26004); English edition: Fractional integrals and derivatives: theory and applications, Gordon and Breach (1992)]. In this paper three types of a ``truncation'' of the Chen-Marchaud constructions and the integral representations for such truncated Chen- Marchaud fractional derivatives are proved. These results and the theorems on approximation of the identity apply to an inversion and a characterization of the Chen fractional integrals of functions locally \(p\)-integrable on the real axis. The results obtained show that the Chen- Marchaud integro-differentiation has an advantage in comparison with the usual Liouville integro-differentiation because of the possibility to consider locally \(p\)-summable functions with an arbitrary behavior at infinity.