Volume 28, Issue 1
Estimates of Linear Relative n-widths in Lp[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 shapepreservingrelative to the cone of all k-times differentiable functions with non-negative k-thderivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the orderof 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 willbe able to find some asymptotic estimates of linear relative n-width of sets of differentiablefunctions in the space L^p[0,1], p ∈ N.

• Keywords

Shape preserving approximation linear n-width

41A35 41A29

@Article{ATA-28-38, author = {Sergei P. Sidorov}, title = {Estimates of Linear Relative n-widths in Lp[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 shapepreservingrelative to the cone of all k-times differentiable functions with non-negative k-thderivative on [0,1], and the operators Ln are assumed to be of finite rank n, then the orderof 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 willbe able to find some asymptotic estimates of linear relative n-width of sets of differentiablefunctions in the space L^p[0,1], p ∈ 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} }