On Estimates of Uniform Approximations by Rational Fourier–Chebyshev Integral Operators for a Certain Choice of Poles

The rational Fourier–Chebyshev integral operator with specially chosen poles is considered on the closed interval $[-1,1]$. With the help of the previously obtained upper bound for the uniform approximations of the functions $|x|^s$, $s>0$, on the closed interval $[-1,1]$ by means of the method of rational approximation in use, an asymptotic representation of the corresponding majorant under some conditions on the poles of the approximating function is obtained. To solve this problem, a method has been developed that is based on the classical Laplace method of studying the asymptotic behavior of integrals. The case of modified “Newman parameters” is studied in detail. The values of these parameters are found for which the highest rate of uniform approximations is ensured. In this case, the orders of uniform rational approximations turn out to be higher than those for the corresponding polynomial analogs.