On one method of rational approximations of Riemann - Liouville type integral on segment

We study rational approximations of functions defined by a Riemann - Liouville integral on the interval $[-1,1]$ with a density belonging to some classes of continuous functions. As the approximation apparatus, the Riemann - Liouville type integral with a density being a rational Fourier - Chebyshev integral operator serves. We find upper bounds for approximations of the Riemann - Liouville type integral with a bounded density, which depends on the poles and the position of a point in the segment. As a separate problem we study of approximations of Riemann - Liouville type integrals with a density being a function with a power singularity. We obtain uniform upper bounds for approximations with a certain majorant that depends on the position of a point in a segment. We find an asymptotic expression for this majorant, which depends on the poles of approximating rational function. We study the case, when the poles are some modifications of the Newman parameters. We find optimal values of the parameters, for which the approximations have the greatest decay rate. The rate of best rational approximations by the considered method is higher in comparison with the corresponding polynomial analogues.