Optimal spline spaces for $L^2$ $n$-width problems with boundary conditions

TL;DR

This paper identifies optimal spline spaces of all degrees for specific boundary-conditioned function classes in $H^r(0,1)$, ensuring minimal approximation error in the $L^2$ norm with uniform knots.

Contribution

It establishes the existence of optimal spline spaces for boundary-conditioned functions in $H^r(0,1)$ across all degrees ≥ r-1 with uniform knot placement.

Findings

Optimal spline spaces exist for three classes of boundary-conditioned functions.

These spline spaces are optimal in the $L^2$ norm.

Optimal spaces are available for all degrees ≥ r-1 with uniform knots.

Abstract

In this paper we show that, with respect to the $L^2$ norm, three classes of functions in $H^r(0,1)$, defined by certain boundary conditions, admit optimal spline spaces of all degrees $\geq r-1$, and all these spline spaces have uniform knots.