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A bound on the $ L\sb{\infty }$-norm of $ L\sb{2}$-approximation by splines in terms of a global mesh ratio

Author: Carl de Boor
Journal: Math. Comp. 30 (1976), 765-771
MSC: Primary 41A15
MathSciNet review: 0425432
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Abstract: Let $ {L_k}f$ denote the least-squares approximation to $ f \in {{\mathbf{L}}_1}$ by splines of order k with knot sequence $ {\mathbf{t}} = ({t_i})_1^{n + k}$. In connection with their work on Galerkin's method for solving differential equations, Douglas, Dupont and Wahlbin have shown that the norm $ {\left\Vert {{L_k}} \right\Vert _\infty }$, of $ {L_k}$ as a map on $ {{\mathbf{L}}_\infty }$ can be bounded as follows,

$\displaystyle {\left\Vert {{L_k}} \right\Vert _\infty } \leqslant {\operatorname{const}_k}{M_{\mathbf{t}}},$

with $ {M_{\mathbf{t}}}$ a global mesh ratio, given by

$\displaystyle {M_{\mathbf{t}}}: = \mathop {\max }\limits_i \;\Delta {t_i}/\min \,\{ \Delta {t_i}\vert\Delta {t_i} > 0\}.$

Using their very nice idea together with some facts about B-splines, it is shown here that even

$\displaystyle \Vert L_k \Vert _\infty \leqslant \operatorname{const}_k(M_{\mathbf{t}}^{(k)})^{1/2} $

with the smaller global mesh ratio $ M_{\mathbf{t}}^{(k)}$ given by

$\displaystyle M_{\mathbf{t}}^{(k)}: = \mathop {\max }\limits_{i,j} ({t_{i + k}} - {t_i})/{t_{j + k}} - {t_j}).$

A mesh independent bound for $ {{\mathbf{L}}_2}$-approximation by continuous piecewise polynomials is also given.

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Keywords: Least-squares approximation by splines, error bounds
Article copyright: © Copyright 1976 American Mathematical Society