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

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 ${\left \| {{L_k}} \right \|_\infty }$, of ${L_k}$ as a map on ${{\mathbf {L}}_\infty }$ can be bounded as follows, \[ {\left \| {{L_k}} \right \|_\infty } \leqslant {\operatorname {const}_k}{M_{\mathbf {t}}},\] with ${M_{\mathbf {t}}}$ a global mesh ratio, given by \[ {M_{\mathbf {t}}}: = \max \limits _i \;\Delta {t_i}/\min \{ \Delta {t_i}|\Delta {t_i} > 0\}.\] Using their very nice idea together with some facts about B-splines, it is shown here that even \[ \| L_k \|_\infty \leqslant \operatorname {const}_k(M_{\mathbf {t}}^{(k)})^{1/2} \] with the smaller global mesh ratio $M_{\mathbf {t}}^{(k)}$ given by \[ M_{\mathbf {t}}^{(k)}: = \max \limits _{i,j} ({t_{i + k}} - {t_i})/{t_{j + k}} - {t_j}).\] A mesh independent bound for ${{\mathbf {L}}_2}$-approximation by continuous piecewise polynomials is also given.