A cubic spline of tri-monotone approximation

Abstract

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 $???? \in N$ ”almost” equidistant knots $????_????$ such that
${‖???? - ????‖}_{[????_????, ????_{????-1}]} ≤ ???? ω_4 (????, (???? - ????) / ????, [????_{????+4}, ????_{????-5}] \cap [????, ????]), ???? = 1,...,????,$
where $????$ is an absolute constant, $????_4 (????,????,[\cdot,\cdot])$ is the 4-th modulus of smoothness of $????$, and ${|| \cdot ||}_{[\cdot, \cdot]}$ is the max-norm.