Construction of an optimal quadrature formula for the approximation of fractional integrals

In this article, the optimal quadrature formula construction is elucidated for numerical integration of the right Riemann-Liouville integral in the Hilbert space of real-valued functions. Firstly, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. Since the error functional is illustrated on the Hilbert space, the quadrature formula that we are constructing is exact for zero element of the space. That is, we have the conditions that the influence of the error functional on these functions is equal to zero. Then, the Lagrange function is constructed to find the conditional extremum of the error functional. Thereby, a linear equations system is gained for the coefficients of the optimal quadrature formula. The existence and uniqueness of the solution of the obtained system are analyzed.