Conjugate Rational Fourier–Chebyshev Operator and Its Approximation Properties

In this paper, a conjugate rational integral Fourier–Chebyshev operator associated with a system of Chebyshev–Markov algebraic fractions is constructed. Pointwise estimates of approximations are obtained on the intercept [–1, 1] of the conjugate function with a density of (1 – x)γ, γ > 1/2, alongside with uniform estimates of approximations expressed in terms of a certain majorant. The asymptotic expression of this majorant and optimal values of parameters providing the highest speed of its descent are found. As a corollary, the corresponding estimates of approximations are given on the intercept [–1, 1] of the studied conjugate function as partial sums of conjugate polynomial Fourier–Chebyshev series.