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, 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.
Highlights
In this paper, we aim to reformulate the well-known theorems on the Riemann–Liouville operator action in terms of the Jacobi series coefficients
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
Our first aim is to reformulate in terms of the Jacoby series coefficients the previously known theorems describing the Riemann–Liouville operator action in the weighted spaces of Lebesgue p-th power integrable functions, and our second aim is to approach a little bit closer to solving the problem of whether the Riemann–Liouville operator acting in the weighted space of Lebesgue square integrable functions is simple
Summary
In this paper, we aim to reformulate the well-known theorems on the Riemann–Liouville operator action in terms of the Jacobi series coefficients. Even though this type of problems was well studied by such mathematicians as Rubin B.S. Our main interest lies in a rather different field of studying: the mapping theorems for the Riemann–Liouville operator via the Jacobi polynomials. This approach gives us the advantage of getting results in terms of the Jacobi series coefficients, as well as the concrete achievements.
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