On approximations of Riemann–Liouville integral on a segement by rational Fourier–Chebyshev integral operators

Approximations of Riemann–Liouville integral on a segment by rational integral operators Fourier–Chebyshev are investigated. An integral representation of the approximations is found. Rational approximations Riemann–Liouville integral with density $\varphi_\gamma(x) = (1-x)^\gamma,$ $\gamma \in (0,+\infty)\backslash\mathbb{N},$ are studied, estimates of pointwise and uniform approximations are established. In the case of one pole in an open complex plane, an asymptotic expression is obtained for the approximating function majorants of uniform approximations and the optimal value of the parameter at which the majorant has the asymptotically highest rate of decrease. As a consequence, estimates of approximations of Riemann–Liouville integral with density belonging to some classes of continuous functions on the segment by partial sums of the polynomial Fourier–Chebyshev series are obtained.