The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties

Pub Date : 2024-01-01 DOI:10.1134/s0037446624010129

引用次数: 0

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 \( s\in(0,2) \) .

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傅里叶-切比雪夫有理积分算子的里兹-齐格蒙德和及其近似性质

摘要通过研究傅里叶-切比雪夫有理积分算子的某个里兹-齐格蒙德和的近似性质，以及对几何上不同极点数目的约束，我们得到了算子的积分表达式。我们找到了函数 \( |x|^{s} \) 在线段 \( [-1,1] \) 上与\( s\in(0,2) \) 的点逼近和均匀逼近的上界，均匀逼近的大数的渐近表达式，以及提供最大大数下降率的逼近参数的最优值。我们分别研究了傅里叶-切比雪夫多项式级数的 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 \( \gamma\in(0,1] \)的近似值，以及函数 \( |x|^{s} \)与 \( s\in(0,2) \)的点和均匀近似值。

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