On convex approximation by quadratic splines (Q1914800)

This note is devoted to giving a short and constructive proof for the existence of a \(C^1\) convex quadratic spline with \(n\) equidistant knots which approximates a convex function \(f\in C[0,1]\) at the order \(\omega_3(f,1/n)\). This improves a result due to \textit{Y. K. Hu} [J. Approximation Theory 74, No. 1, 69-82 (1993; Zbl 0789.41005)], where the construction was much more complicated and the knots where only ``basically equidistant''. The authors end their note with a modified construction of a \(C^2\) convex cubic spline which provides the same degree of approximation. The ideas of this construction have since been extended by \textit{R. A. DeVore}, \textit{Y. K. Hu} and the reviewer [Constructive Approximation 12, No. 3, 409-422 (1996)] to convex approximation in the \(L_p\) norm.