On best constants in $L^2$ approximation

TL;DR

This paper establishes bounds on $L^2$ approximation constants, introduces a superconvergent numerical method for their computation, and explores implications for optimal spline spaces in Isogeometric Analysis.

Contribution

It provides explicit bounds, a superconvergent numerical method, and insights into the asymptotic behavior of $L^2$ $n$-widths, with applications to optimal spline spaces.

Findings

Explicit bounds on $L^2$ $n$-widths.

A superconvergent numerical computation method.

Formulation of a conjecture on asymptotic behavior.

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.