One cubic 3-monotone spline

TL;DR

This paper constructs cubic 3-monotone splines that closely approximate 3-monotone functions with explicit error bounds based on the function's smoothness, using nearly equidistant knots.

Contribution

It introduces a method to construct cubic 3-monotone splines with explicit approximation error bounds for 3-monotone functions.

Findings

Spline approximation error is bounded by the 4th modulus of smoothness.

Constructed splines have nearly equidistant knots.

Approximation is valid on each subinterval with a universal constant.

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 $n\in \Bbb N$ "almost" equidistant knots $a_j$ such that $$ \left\Vert f-s \right\Vert_{[a_j,a_{j-1}]} \le c\, \omega_4 \left(f,(b-a)/n,[a_{j+4},a_{j-5}]\cap [a,b]\right), \quad j=1,...,n, $$ 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.