Shape-preserving kolmogorov widths of classes of s-monotone integrable functions - Konovalov, V. N. (Ukrains’kyi Matematychnyi Zhurnal, 2004)
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 $Δ^s_{+} B_p := Δ^s_{+} \bigcap B_p,\; 1 ≤ p ≤ ∞$, where $B_p$ is the unitball of the space $L_p$, and let $Δ^s_{+} L_p := Δ^s_{+} \bigcap L_p,\; 1 ≤ q ≤ ∞$. For every $s ≥ 3$ and $1 ≤ q ≤ p ≤ ∞$, exact orders of the shape-preserving Kolmogorov widths$$d_n (Δ^s_{+} B_p, Δ^s_{+} L_p )_{L_p}^{\text{kol}} := \inf_{M^n \in \mathcal{M}^n} \sup_{x \in Δ^s_{+} B_p} \inf_{y \in M^n \bigcap Δ^s_{+} L_p} ||x - y||_{L_p},$$are obtained, where $\mathcal{M}^n$ is the set of all affine linear manifolds $M^n$ in $L_q$ such that $\dim М^n ≤ n$ and $M^n \bigcap Δ^s_{+} L_p eq \emptyset$.