A bound on the 𝐿_{∞}-norm of 𝐿₂-approximation by splines in terms of a global mesh ratio

Abstract

Let L k f {L_k}f denote the least-squares approximation to f ∈ L 1 f \in {{\mathbf {L}}_1} by splines of order k with knot sequence t = ( t i ) 1 n + k {\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 ‖ L k ‖ ∞ {\left \| {{L_k}} \right \|_\infty } , of L k {L_k} as a map on L ∞ {{\mathbf {L}}_\infty } can be bounded as follows, \[ ‖ L k ‖ ∞ ⩽ const k M t , {\left \| {{L_k}} \right \|_\infty } \leqslant {\operatorname {const}_k}{M_{\mathbf {t}}}, \] with M t {M_{\mathbf {t}}} a global mesh ratio, given by \[ M t := max i Δ t i / min { Δ t i | Δ t i > 0 } . {M_{\mathbf {t}}}: = \max \limits _i \;\Delta {t_i}/\min \,\{ \Delta {t_i}|\Delta {t_i} > 0\}. \] Using their very nice idea together with some facts about B -splines, it is shown here that even \[ ‖ L k ‖ ∞ ⩽ const k ⁡ ( M t ( k ) ) 1 / 2 \| L_k \|_\infty \leqslant \operatorname {const}_k(M_{\mathbf {t}}^{(k)})^{1/2} \] with the smaller global mesh ratio M t ( k ) M_{\mathbf {t}}^{(k)} given by \[ M t ( k ) := max i , j ( t i + k − t i ) / t j + k − t j ) . M_{\mathbf {t}}^{(k)}: = \max \limits _{i,j} ({t_{i + k}} - {t_i})/{t_{j + k}} - {t_j}). \] A mesh independent bound for L 2 {{\mathbf {L}}_2} -approximation by continuous piecewise polynomials is also given.