EFFICIENT L2 APPROXIMATION BY SPLINES

Let SNk(t) be the linear space of k-th order splines on [0, 1] having the simple knots ti determined from a fixed function t by the rule ti=t(i/N). In this paper we introduce sequences of operators {QN}N∞=1 from Ck[0, 1] to SNk(t) which are computationally simple and which, as N→∞, give essentially the best possible approximations to f and its first k-1 derivatives, in the norm of L2[0, 1]. Precisely, we show that Nk-1({norm of matrix}(f-QNf)i{norm of matrix}-dist2(f(1), SNk-1(t)))→0 for i=0, 1, ..., k-1. Several numerical examples are given. © 1979 Springer-Verlag.