The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties

Studying the approximation properties ofa certain Riesz–Zygmund sumof Fourier–Chebyshev rational integral operatorswith constraints on the number of geometrically distinct poles,we obtain an integral expression of the operators.We find upper bounds for pointwise and uniform approximationsto 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 approximantproviding the greatest decrease rate of the majorant.We separately studythe approximation properties ofthe 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) $ .