Approximations of conjugate functions by partial sums of conjugate Fourier series with respect to a certain system of Chebyshev – Markov algebraic fractions

We investigate approximative properties of partial sums of conjugate Fourier series with respect to one system of Chebyshev – Markov algebraic fractions. The main results of previously known works on approximations of conjugate functions in polynomial and rational cases are presented. One system of algebraic fractions Chebyshev – Markov is introduced and the construction of the conjugate rational Fourier – Chebyshev series corresponding to it is carried out. An integral representation of the conjugate function approximations by partial sums of the constructed conjugate series is found. The approximation of functions conjugate to $|x|^s, 1 < s < 2,$ on the interval $[-1,1]$ by partial sums of conjugate rational Fourier – Chebyshev series is studied. The integral representation of approximations, estimates of approximations by the studied method depending on the position of the point $x$ on the interval, and their asymptotic expressions for $n \to \infty$ are found. The optimal value of the parameter at which the deviation of partial sums of the conjugate rational Fourier – Chebyshev series from the conjugate function $|x|^s, 1 < s < 2,$ on the interval $[-1,1]$ have the highest rate of tendency to zero is established. As a consequence of the results obtained, the problem of approximations of a function conjugate to $|x|^s, s > 1,$ by partial sums of the conjugate Fourier series on the Chebyshev polynomial system of the first kind is studied in detail.