(untitled) (Transactions of Institute of Mathematics, the NAS of Ukraine, 2017)
For any 3-monotone on [????,????] function ???? (its third divided differences are nonnegative for all choices of four distinct points, or equivalently, ???? has a convex derivative on (????,????)) we construct a cubic 3-monotone (like ????) spline ???? with ???? N ”almost” equidistant knots ????_???? such that ‖???? - ????‖_[????_????, ????_????-1] ≤ ???? ω_4 (????, (???? - ????) / ????, [????_????+4, ????_????-5] [????, ????]), ???? = 1,...,????, where ???? is an absolute constant, ????_4 (????,????,[·,·]) is the 4-th modulus of smoothness of ????, and || · ||_[·, ·] is the max-norm.