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.

在本文中，我们研究了共轭付里叶级数的部分和与某些切比雪夫-马尔可夫代数系统的近似性质。我们列举了在研究多项式和有理情况下的共轭函数近似值的已知工作中获得的主要结果。我们介绍了Chebyshev-Markov代数分数的系统，并构造了相应的共轭有理Fourier-Chebyshev级数。我们通过构造的共轭系列的部分和获得共轭函数近似的积分表示。此外，我们研究函数的近似值，该函数在段\（[-1,1]，\）上与\（| x | ^ s，1 <s <2，\）共轭由共轭有理傅里叶-Chebyshev级数的部分和得出。我们获得近似值的积分表示，使用考虑的方法根据点x在线段上的位置建立它们的估计，并使用\（n \ to \ infty \）找到它们的渐近形式。我们还计算了参数的最佳值，该参数使共轭有理傅里叶– Chebyshev级数的部分和与共轭函数\（| x | ^ s，1 <s <2，\）的偏差趋于零。 （[-1,1] \）以最高的速率。获得的结果使我们能够彻底研究与\（| x | ^ s，s> 1，\）共轭函数的逼近性质。 关于第一类切比雪夫多项式系统的共轭傅里叶级数的部分和。