EfficientL 2 approximation by splines

LetS_N^k(t) be the linear space ofk-th order splines on [0, 1] having the simple knotst_i determined from a fixed functiont by the rulet_i=t(i/N). In this paper we introduce sequences of operators {Q_N}_N=1 fromC^k[0, 1] toS_N^k(t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL₂[0, 1]. Precisely, we show thatN^k–1((f–Q_Nf)ⁱ–dist₂(f⁽¹⁾,S_N^k–1(t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816