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 I, the corresponding divided differences [x; t_0,..., t_s ] of order s are nonnegative. Let Δ^s_+ B_p := Δ^s_+ B_p, 1 ≤ p ≤ ∞, where B_p is the unit ball of the space L_p, and let Δ^s_+ L_p := Δ^s_+ 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^kol := _M^n M^n _x Δ^s_+ B_p _y M^n Δ^s_+ L_p ||x - y||_L_p, are obtained, where M^n is the set of all affine linear manifolds M^n in L_q such that М^n ≤ n and M^n Δ^s_+ L_p eq .