Approximations on the closed interval \([-1,1]\) of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression for the majorant of uniform approximations and an optimal parameter value ensuring the greatest rate of approximation by the method used in the paper are found. In the case of the even multiplicity of the poles of the approximating function, the asymptotic estimate is sharp. Examples of approximations of concrete functions are given.

中文翻译：

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Chebyshev-Markov系统上傅里叶级数部分和对Markov函数的有理逼近

考虑在Chebyshev-Markov有理分数系统上傅立叶级数的部分和对经典马尔可夫函数的组合的封闭区间\（[-1,1] \）的近似。建立近似的逐点和均匀估计。对于测度的导数弱等于幂函数的情况，通过本文中使用的方法，找到了统一逼近的主观的渐近表达式和确保最大逼近率的最优参数值。在逼近函数极点的偶数倍的情况下，渐近估计很尖锐。给出了具体函数的近似示例。