Conjugate rational Foutier–Chebyshev operator and its approximation properties

In this paper, a conjugate rational integral Fourier–Chebyshev operator associated with the system of Chebyshev–Markov algebraic fractions is constructed. Pointwise estimates of approximations on the segment $[-1,~1]$ of the conjugate function with density $(1-x)^{\gamma}, \gamma>1/2,$ and uniform estimates of approximations expressed in terms of a certain majorant are obtained. Asymptotic expression for majorants of approximations and optimal values of parameters that provide the highest speed decreasing majorants are found. At the corollary the corresponding estimates of approximations on the segment $[-1,~1]$ of the conjugate function under study by partial sums of conjugate polynomial Fourier–Chebyshev series are given.