Shape-preserving kolmogorov widths of classes of <em class="a-plus-plus">s</em>-monotone integrable functions

Abstract

Let $s ∈ ℕ$ and $Δ^s_{+}$ be a set of functions $x$ which are defined on a finite interval $I$ and are such that, for all collections of $s + 1$ pairwise different points $t_0,..., t_s \in I$, the corresponding divided differences $[x; t_0,..., t_s ]$ of order $s$ are nonnegative. Let $\Delta^s_{+} B_p := \Delta^s_{+} \bigcap B_p,\; 1 \leq p \leq \infty$, where $B_p$ is the unit ball of the space $L_p$, and let $\Delta^s_{+} L_p := \Delta^s_{+} \bigcap L_p,\; 1 \leq q \leq \infty$. For every $s \geq 3$ and $1 \leq q \leq p \leq \infty$, exact orders of the shape-preserving Kolmogorov widths $$d_n (\Delta^s_{+} B_p, \Delta^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in \Delta^s_{+} B_p} \inf_{y \in M^n \bigcap \Delta^s_{+} L_p} ||x - y||_{L_p},$$ are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n \leq n$ and $M^n \bigcap \Delta^s_{+} L_p \neq \emptyset$.