Approximations on classes of Poisson integrals by Fourier–Chebyshev rational integral operators

Introducing some classes of the functions defined by Poisson integrals on the segment $[-1,1]$ and studying approximations by Fourier–Chebyshev rational integral operators on the classes, we establish integral expressions for approximations and upper bounds for uniform approximations. In the case of boundary functions with a power singularity on $[-1,1]$, we find the upper bounds for pointwise and uniform approximations and an asymptotic expression for a majorant of uniform approximations in terms of rational functions with a fixed number of prescribed geometrically distinct poles. Considering two geometrically distinct poles of the approximant of even multiplicity, we obtain asymptotic estimates for the best uniform approximation by this method with a higher convergence rate than polynomial analogs.