Приближения сопряженных функций частичными суммами сопряженных рядов Фурье по одной системе алгебраических дробей Чебышева-Маркова

Abstract

In this paper, we study approximative properties of partial sums of a conjugate Fourier series with respect to a certain system of Chebyshev – Markov algebraic fractions. We cite the main results obtained in known works devoted to studying approximations of conjugate functions in polynomial and rational cases. We introduce a system of Chebyshev – Markov algebraic fractions and construct the corresponding conjugate rational Fourier – Chebyshev series. We obtain an integral representation for approximations of the conjugate function by partial sums of the constructed conjugate series. Moreover, we study approximations of the function, which is conjugate to $|x|^s, 1 < s < 2,$ on the segment $[-1,1],$ by partial sums of the conjugate rational Fourier – Chebyshev series. We obtain an integral representation of approximations, establish their estimates, using the considered method, in dependence of the location of the point x on the segment, and find their asymptotic forms with $n \to \infty$ . We also calculate the optimal value of the parameter that makes deviations of partial sums of the conjugate rational Fourier – Chebyshev series from the conjugate function $|x|^s, 1 < s < 2,$ tend to zero on the segment $[-1,1]$ at the highest possible rate. The obtained results have allowed us to thoroughly study properties of approximations of the function conjugate to $|x|^s, s > 1,$ by partial sums of the conjugate Fourier series with respect to a system of Chebyshev polynomials of the first kind.