Abstract

Studying the approximation properties of a certain Riesz–Zygmund sum of Fourier–Chebyshev rational integral operators with constraints on the number of geometrically distinct poles, we obtain an integral expression of the operators. We find upper bounds for pointwise and uniform approximations to the function \( |x|^{s} \) with \( s\in(0,2) \) on the segment \( [-1,1] \) , an asymptotic expression for the majorant of uniform approximations, and the optimal values of the parameter of the approximant providing the greatest decrease rate of the majorant. We separately study the approximation properties of the Riesz–Zygmund sums for Fourier–Chebyshev polynomial series, establish an asymptotic expression for the Lebesgue constants, and estimate approximations to \( f\in H^{(\gamma)}[-1,1] \) and \( \gamma\in(0,1] \) as well as pointwise and uniform approximations to the function  \( |x|^{s} \) with \( s\in(0,2) \) .

中文翻译：

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傅里叶-切比雪夫有理积分算子的Riesz-Zygmund和及其逼近性质

摘要

研究傅里叶-切比雪夫有理积分算子的某个 Riesz-Zygmund 和的近似性质，并限制几何上不同的极点数量，我们得到了算子的积分表达式。我们找到函数\( |x|^{s} \)的逐点和均匀近似的上限，其中\( s\in(0,2) \)在段\( [-1,1] \)上，一致近似的主函数的渐近表达式，以及提供主函数最大下降率的近似参数的最佳值。我们分别研究了 Fourier-Chebyshev 多项式级数的 Riesz-Zygmund 和的近似性质，建立了 Lebesgue 常数的渐近表达式，并估计了\( f\in H^{(\gamma)}[-1,1 ] \)和\( \gamma\in(0,1] \)以及函数 \( |x|^{s} \)和\( s\in(0,2) \ 的逐点和均匀近似）。