On rational approximations of conjugate function on an interval by conjugate Vallée Poussin sums

The approximations of the conjugate function on the segment $[-1, 1]$ by Vallée Poussin 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, the estimation of pointwise approximations and uniform approximations with a certain majorant are obtained for a conjugate function with density $(1-x)^\gamma$, $\gamma\in(0,1)$. Its asymptotic expression for $n\to\infty$, depending on the parameters of the approximating function, is established. The optimal values of the parameters at which the highest rate of decreasing majorant is provided are found. As a consequence, the estimates of approximations of conjugate function on the segment $[-1, 1]$ by Vallée Poussin sums of conjugate polynomial Fourier – Chebyshev series are found.