A Fejér rational integral operator on a closed interval and approximation of functions with a power-law singularity

Rational approximations of continuous functions and functions with a power-law singularity on a closed interval are studied by means of integral Fejér-type operators. Upper estimates of approximations of continuous functions on a closed interval are derived; the estimates are expressed in terms of the modulus of continuity and depend on the position of a point in the interval. Rational approximations of the function $(1-x)^\gamma$, $\gamma\in (0,1)$, on the interval $[-1,1]$ are studied. Upper estimates of uniform approximations in terms of the corresponding majorant and an asymptotic expression as $n\to\infty$ of this majorant are found. In the case of a fixed number of poles of the approximating function, optimal values of the parameters are obtained, for which the majorant of the uniform approximations decreases at the highest rate. A consequence of the results obtained is asymptotic estimates of approximations of some specific functions by Fejér sums of polynomial Fourier–Chebyshev series.