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

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.

On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

“…From now on, contrary to Reference [11], we consider the right-hand side case, assuming that α ∈ (−1, 0), but the reasonings corresponding to the right-hand side case are absolutely analogous.…”

“…The following theorem is the very mapping theorem (see Reference [11]) formulated in terms of the Jacobi series coefficients. Here we give the modified form corresponding to the right-hand side case.…”

“…We also need an adopted version (see [11]) of the Zigmund-Marczinkevich theorem (see Reference [26]), which establishes the following.…”

“…The main disadvantage of the results presented in Reference [10] consists of some gaps of the parameter values in the conditions; moreover, the notorious problem related to p = 1/α (the detailed information can be found in paragraph 3.3 [10], p. 91) remained completely unsolvable. All these create the prerequisite to invent another approach for studying the Riemann-Liouville operator action that was successfully investigated in Reference [11], and below we write out some of its highlights. Despite the fact that the idea of using the Jacobi polynomials is not novel and many papers have been devoted to this topic [12][13][14][15][16][17][18], we confirm the main advantage of the method, used in Reference [11] and based on the results [19][20][21][22][23], which are still relevant and allow us to obtain some interesting results.…”

“…The main challenge of this paper is to improve and clarify the results of Reference [11]. In particular we need to find a simple condition, imposed on the right-hand side of the Abel equation, under which Theorem 2 [11] is applicable. For this purpose we make an attempt to solve this problem by using absolute convergence of a series.…”