(untitled) (Analysis in Theory and Applications, 2012)
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 N.