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 - <p style="text-align: justify;">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 $L^p[0,1], p \in N$.</p>