Riemann-Liouville Operator in Weighted L_p Spaces via the Jacobi Series Expansion

TL;DR

This paper employs Jacobi polynomial expansions to analyze Riemann-Liouville fractional operators in weighted Lp spaces, extending fractional calculus theory with new conditions and invariant subspace results.

Contribution

It introduces a novel approach using Jacobi series to study fractional integral and derivative operators, providing new reformulations and invariant subspace insights.

Findings

Derived a theorem characterizing fractional integral operator action via Jacobi series coefficients.

Established a sufficient condition for representing functions through fractional integrals in Jacobi series.

Identified invariant subspaces of the Riemann-Liouville operator in weighted Lp spaces.

Abstract

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 what gives us an 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.