The de la Vallée Poussin sums of Fourier–Chebyshev rational integral operators and approximations to Poisson integrals on the segment

We study the rational approximations of the functions represented by Poisson integrals on the segment $[-1,1]$ with constraints on the number of geometrically distinct poles of the approximant. As the approximation method we choose the de la Vallée Poussin sums of Fourier–Chebyshev rational integral operators. For the method of rational approximation we establish integral representations for approximations and upper bounds for uniform approximations on the classes of Poisson integrals on $[-1,1]$. We consider the classes of Poisson integrals whose boundary function has a power singularity on $[-1,1]$. In this case we find upper bounds for pointwise and uniform approximations and an asymptotic expression for a majorant of uniform approximation. We study approximations by the rational de la Vallée Poussin sums with two geometrically distinct poles and establish the values of the parameters ensuring the best uniform rational approximations by this method. We show that in this case the majorants of the best uniform approximations decay faster than the corresponding polynomial analogs. As a corollary, we consider the approximations of the functions defined by Poisson integrals on the segment by the de la Vallée Poussin sums of the Fourier–Chebyshev polynomial series.