On rational conjugate Fejér sums on an interval and approximations of the conjugate function

The approximations of the conjugate function on the segment $[-1, 1]$ by Fejér sums of conjugate rational integral Fourier – Chebyshev operators with restrictions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established. An integral representation of approximations, estimation of pointwise approximations and uniform approximations with a certain majorant are obtained for a conjugate function with density $(1-x)^\gamma$, $\gamma\in(1/2,1)$. Its asymptotic expression for $n\to\infty$, depending on the parameters of the approximating function is established. In the final part, the optimal values of parameters at which the highest rate of decreasing majorant is provided are found. As a corollary, the estimates of approximations of conjugate function on the segment $[-1, 1]$ by Fejér sums conjugate polynomial Fourier – Chebyshev series are found.