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Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion

Kabardino-Balkarian Scientific Center, RAS, Nalchik 360002, Russia
Axioms 2019, 8(2), 75; https://doi.org/10.3390/axioms8020075
Received: 20 April 2019 / Revised: 10 June 2019 / Accepted: 18 June 2019 / Published: 23 June 2019
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces. View Full-Text
Keywords: fractional derivative; fractional integral; Riemann–Liouville operator; Jacobi polynomials; Legendre polynomials; invariant subspace fractional derivative; fractional integral; Riemann–Liouville operator; Jacobi polynomials; Legendre polynomials; invariant subspace
MDPI and ACS Style

Kukushkin, M.V. Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion. Axioms 2019, 8, 75.

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