On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix of Jacobi nodes,Izvestiya: Mathematics

当前位置： X-MOL 学术 › Izv. Math. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix of Jacobi nodes
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2020-12-30 , DOI: 10.1070/im8992
A. Yu. Trynin ₁

Affiliation

Let sequences $\{\alpha_n\}_{n=1}^{\infty}$ , $\{\beta_n\}_{n=1}^{\infty}$ satisfy the relations $\alpha_n\in \mathbb{R}$ , $\beta_n\in\mathbb{R}$ , $\alpha_n=o(\sqrt{n/\ln n})$ , $\beta_n=o(\sqrt{n/\ln n})$ as $n\to \infty$ , and let $[a,b]\subset (0,\pi)$ and $f\in C[a,b]$ . We redefine the function $f$ as $F$ on the interval $[0,\pi]$ by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function $f$ and the pair of sequences $\{\alpha_n\}_{n=1}^{\infty}$ , $\{\beta_n\}_{n=1}^{\infty}$ be connected by the equiconvergence condition. Then for the classical Lagrange–Jacobi interpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ to approximate $f$ uniformly with respect to $\theta$ on $[a,b]$ it is sufficient that $f$ have bounded variation $V^{b}_{a}(f)< \infty$ on $[a,b]$ . In particular, if the sequences $\{\alpha_n\}_{n=1}^{\infty}$ and $\{\beta_n\}_{n=1}^{\infty}$ are bounded, then for the classical Lagrange–Jacobi interpolation processes $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ to approximate $f$ uniformly with respect to $\theta$ on $[a,b]$ it is sufficient that the variation of $f$ be bounded on $[a,b]$ , $V^{b}_{a}(f)<\infty$ .

中文翻译：

用带雅可比节点矩阵的拉格朗日插值多项式对有界变差函数的均匀逼近

让序列 $\{\alpha_n\}_{n=1}^{\infty}$ ， $\{\beta_n\}_{n=1}^{\infty}$ 满足关系 $\alpha_n\in \mathbb{R}$ ， $\beta_n\in\mathbb{R}$ ， $\alpha_n=o(\sqrt{n/\ln n})$ ， $\beta_n=o(\sqrt{n/\ln n})$ 如 $n\to \infty$ ，让 $[a,b]\subset (0,\pi)$ 与 $f\in C[a,b]$ 。我们重新定义函数 $f$ 作为 $F$ 在区间 $[0,\pi]$ 通过以这样的方式多角弧，该函数保持连续和消失的区间的端部的附近。也让功能 $f$ 和序列对 $\{\alpha_n\}_{n=1}^{\infty}$ ， $\{\beta_n\}_{n=1}^{\infty}$ 由equiconvergence条件进行连接。然后，对于经典的拉格朗日-雅可比内插处理 $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ 来近似 $f$ 相对于均匀地 $\theta$ 对 $[a,b]$ 它是足够 $f$ 有界变差 $V^{b}_{a}(f)< \infty$ 上 $[a,b]$ 。特别地，如果序列 $\{\alpha_n\}_{n=1}^{\infty}$ 和 $\{\beta_n\}_{n=1}^{\infty}$ 是有界的，那么对于传统的拉格朗日-雅可比插值处理 $\mathcal{L}_n^{(\alpha_n,\beta_n)}(F,\cos\theta)$ 来近似 $f$ 相对于均匀地 $\theta$ 对 $[a,b]$ 就足够了的变化 $f$ 来界定上 $[a,b]$ ， $V^{b}_{a}(f)<\infty$ 。

更新日期：2020-12-30

点击分享查看原文

点击收藏

阅读更多本刊最新论文

全部期刊列表>>