![Estimates of Linear Relative $n$-widths in $L^p[0, 1]$](https://global-sci.com/images/articles/74053.png?w=900&h=&fit=&c=2&assets_cache=20241002154208&bg=white)
Year: 2012
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 1 : pp. 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 $L^p[0,1], p \in N$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2012.v28.n1.5
Analysis in Theory and Applications, Vol. 28 (2012), Iss. 1 : pp. 38–48
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 11