The Riesz–Zygmund sums of Fourier–Chebyshev rational integral operators and their approximation properties

Studying the approximation properties of a certain Riesz–Zygmund sum of Fourier–Chebyshev rational integral operators with constraints on the number of geometrically distinct poles, we obtain an integral expression of the operators. We find upper bounds for pointwise and uniform approximations to the function $|x|^s$ with $s \in (0, 2)$ on the segment $[-1,1]$, an asymptotic expression for the majorant of uniform approximations, and the optimal values of the parameter of the approximant providing the greatest decrease rate of the majorant. We separately study the approximation properties of the Riesz–Zygmund sums for Fourier–Chebyshev polynomial series, establish an asymptotic expression for the Lebesgue constants, and estimate approximations to $f \in H^{(\gamma)}[-1,1]$ and $\gamma \in (0, 1]$ as well as pointwise and uniform approximations to the function $|x|^s$ with $s \in (0, 2)$.