Mapping Theorems for a Convolution Operator via the Jacoby Polynomials

In this report 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 modications 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.