On best constants in $L^2$ approximation

Abstract

In this paper we provide explicit upper and lower bounds on certain $L^2$ $n$-widths, i.e., best constants in $L^2$ approximation. We further describe a numerical method to compute these $n$-widths approximately, and prove that this method is superconvergent. Based on our numerical results we formulate a conjecture on the asymptotic behaviour of the $n$-widths. Finally we describe how the numerical method can be used to compute the breakpoints of the optimal spline spaces of Melkman and Micchelli, which have recently received renewed attention in the field of Isogeometric Analysis.

arXiv comment15 pages, 4 figures and 1 table. Improved the presentation. Article now published in IMAJNA

References in corpus (3)

• Sharp error estimates for spline approximation: explicit constants, $n$-widths, and eigenfunction convergence
• Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion
• Optimal spline spaces for $L^2$ $n$-width problems with boundary conditions

Cited by in corpus (1)

• Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations