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

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.

The autonomous intelligence challenge

The considerable development of modern technology during last decades has been accom-panied by improvements in associated adapted machines’ performance. The improvement process has always been guided by the same rules of researching higher efficiency and more secure effects each time. This leads to a progressive transfer of human action to more adapted and more specific effecting objects, from simple tools for elementary actions to more sophisticated machines for quite complex tasks. Each step of this transfer of human operator action has been realized by delegating to the effecting machine efficiency, accuracy, power and safety—basically all of technical nature and linked to power flux, with the human operator still keeping the mastery of the action to fulfil his own goals.