Shape preserving widths of Sobolev-type classes ofs-monotone functions on a finite interval

LetI be a finite interval andr ∈ ℕ. Denote by △+sLq the subset of all functionsy ∈Lq such that thes-difference △Tsy(·) is nonnegative onI, ∀τ>0. Further, denote by △+sWpr the class of functionsx onI with the seminorm ‖x(r)‖Lp≤1, such that △Tsx≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$d_n \left( {\Delta _ + ^s W_p^r ,\Delta _ + ^s L_q } \right) = \begin{array}{*{20}c} {\inf } \\ {M^n \in M^n } \\ \end{array} \begin{array}{*{20}c} {\sup } \\ {x \in \Delta _ + ^s W_p^r } \\ \end{array} \begin{array}{*{20}c} {\inf } \\ {y \in M^n \cap \Delta _ + ^s L_q } \\ \end{array} \left\| {x - y} \right\|L_q $$ \end{document}, whereMn is the set of all linear manifoldsMn inLq, dimMn≤n, such thatMn ⋂△+sLq ≠ 0.