Shape-Preserving Smoothing of 3-Convex Splines of Degree 4

Abstract

For every 3-convex piecewise-polynomial function s of degree ≤ 4 with n equidistant knots on [0, 1] we construct a 3-convex spline $s_1 (s_1 ∈ C (3))$ of degree ≤ 4 with the same knots that satisfies the inequality $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ where $c$ is an absolute constant and $ω_5$ is the modulus of smoothness of the fifth order.