One cubic 3-monotone spline

Abstract

For any 3-monotone on $[a,b]$ function $f$ (its third divided differences are nonnegative for all choices of four distinct points, or equivalently, $f$ has a convex derivative on $(a,b)$) we construct a cubic 3-monotone (like $f$) spline $s$ with $n\in \mathbb{N}$ "almost" equidistant knots $a_j$ such that $${\Vert f-s\Vert }_{[a_j,a_{j-1}]}\le c{\omega }_4\left(f,(b-a)/n,[a_{j+4},a_{j-5}]\cap [a,b]\right),j=1,...,n,$$ where $c$ is an absolute constant, ${\omega }_4\left(f,t,[\cdot ,\cdot ]\right)$ is the $4$-th modulus of smoothness of $f$, and $||\cdot |{|}_{[\cdot ,\cdot ]}$ is the max-norm.

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