Riemann-Liouville operator in weighted Lp spaces via the Jacoby series expansion

Axioms. 2019. Vol. 8. No. 2. Article 75.

Maksim V. Kukushkin

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.

Keywords: Operator theory, Abel differential equations