On the Orders of Nonlinear Approximations for Classes of Functions of Given Form

Suppose that $\Delta^s_+$ is the set of functions $x\colon I\to\mathbb R$ on a finite interval $I$ such that the divided differences $[x;t_0,\dots,t_s]$ of order $s\in\mathbb N$ of these functions are nonnegative for all collections from $(s+1)$ different points $t_0,\dots,t_s\in I$. For all $s\in\mathbb N$ and $1\le p\le\infty$, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of $L_p$ for classes $\Delta^s_+B_p:=\Delta^s_+\cap B_p$, where $B_p$ is the unit ball in $L_p$. We also establish the asymptotics of pseudodimensional widths in $L_p$ of these classes of functions.