Volume 28, Issue 1
Estimates of Linear Relative $n$-widths in $L^p[0, 1]$

Anal. Theory Appl., 28 (2012), pp. 38-48.

Published online: 2012-03

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• Abstract

In this paper we will show that if an approximation process $\{L_n\}_{n∈N}$ is shape-preserving relative to the cone of all $k$-times differentiable functions with non-negative $k$-th derivative on [0,1], and the operators $L_n$ are assumed to be of finite rank $n$, then the order of convergence of $D^kL_n f$ to $D^k f$ cannot be better than $n^{−2}$ even for the functions $x^k$, $x^{k+1}$, $x^{k+2}$ on any subset of [0,1] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative $n$-width of sets of differentiable functions in the space $L^p[0,1], p \in N$.

• Keywords

Shape preserving approximation, linear $n$-width.

41A35, 41A29

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@Article{ATA-28-38, author = {}, title = {Estimates of Linear Relative $n$-widths in $L^p[0, 1]$}, journal = {Analysis in Theory and Applications}, year = {2012}, volume = {28}, number = {1}, pages = {38--48}, abstract = {

In this paper we will show that if an approximation process $\{L_n\}_{n∈N}$ is shape-preserving relative to the cone of all $k$-times differentiable functions with non-negative $k$-th derivative on [0,1], and the operators $L_n$ are assumed to be of finite rank $n$, then the order of convergence of $D^kL_n f$ to $D^k f$ cannot be better than $n^{−2}$ even for the functions $x^k$, $x^{k+1}$, $x^{k+2}$ on any subset of [0,1] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative $n$-width of sets of differentiable functions in the space $L^p[0,1], p \in N$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2012.v28.n1.5}, url = {http://global-sci.org/intro/article_detail/ata/4539.html} }
TY - JOUR T1 - Estimates of Linear Relative $n$-widths in $L^p[0, 1]$ JO - Analysis in Theory and Applications VL - 1 SP - 38 EP - 48 PY - 2012 DA - 2012/03 SN - 28 DO - http://doi.org/10.4208/ata.2012.v28.n1.5 UR - https://global-sci.org/intro/article_detail/ata/4539.html KW - Shape preserving approximation, linear $n$-width. AB -

In this paper we will show that if an approximation process $\{L_n\}_{n∈N}$ is shape-preserving relative to the cone of all $k$-times differentiable functions with non-negative $k$-th derivative on [0,1], and the operators $L_n$ are assumed to be of finite rank $n$, then the order of convergence of $D^kL_n f$ to $D^k f$ cannot be better than $n^{−2}$ even for the functions $x^k$, $x^{k+1}$, $x^{k+2}$ on any subset of [0,1] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative $n$-width of sets of differentiable functions in the space $L^p[0,1], p \in N$.

Sergei P. Sidorov. (1970). Estimates of Linear Relative $n$-widths in $L^p[0, 1]$. Analysis in Theory and Applications. 28 (1). 38-48. doi:10.4208/ata.2012.v28.n1.5
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