An Application of Spline Approximation with Variable Knots to Optimal Estimation of the Derivative

Abstract

In studying the optimal $L_p$ estimation of $f^\prime$ at $x_0 = t_{n + k} $ from the data $f(t_1 ), \cdots ,f(t_{n + k - 1} )$, where $t_1 < \cdots < t_{n + k - 1} = t_{n + k} $, one naturally arrives at the problem of best $L_q$ approximation of $N_n,k$ from the span of $N_{i,k} ,i = 1, \cdots ,n - 1$, Where $\{ N_{i,k} \} _{i = 1}^n $ are the normalized B-splines with the knot sequence $\{ t_i \} _{i = 1}^{n + k} $ and ${1 / p} + {1 / q} = 1$. We prove that the $L_q$ error $d_q (c({\bf t})N_{n,k} ,{\operatorname{Sp}}\{ N_{i,k} \} _{i = 1}^{n - 1} )$, where $(c({\bf t}) = {1 / {(k - 1)!\Pi _{i = 1}^{k - 2} }}(1 - t_{n + i} )$, is nonincreasing in ${\bf t}$ for $1 \leqq q \leqq \infty $, and actually is decreasing for $1 \leqq q \leqq \infty $, as the knots $t_1 , \cdots ,t_{n + k - 2} $ are moved to the right toward $t_{n + k - 1} $. This agrees with the general conjecture of G. G. Lorentz, namely, “like best approximates like.” Consequently, in approximating $f'(x_0 )$, it is advisable to take the nodes $t_1 , \cdots ,t_{n + k - 2} $ as close to $x_0$ as possible as long as the process is stable. However, it is also shown that this conclusion is no longer valid if one approximates $f'(x_0 ) - \alpha f'(x_0 )$, say for $\alpha = 10$ and $k = 3$.