Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation

TL;DR

This paper derives the precise asymptotic behavior of the best asymmetric Lp-error when approximating twice-differentiable functions with nonnegative Hessian by piecewise linear splines over triangulations, as the number of elements grows large.

Contribution

It provides the exact asymptotics of the optimal asymmetric Lp-error for spline approximation of functions with nonnegative Hessian, a novel result in approximation theory.

Findings

Exact asymptotic formula for the approximation error as N→∞

Characterization of optimal triangulations for approximation

Extension of classical symmetric approximation results to asymmetric case

Abstract

In this paper we study the best asymmetric (sometimes also called penalized or sign-sensitive) approximation in the metrics of the space $L_p$, $1\leqslant p\leqslant\infty$, of functions $f\in C^2\left([0,1]^2\right)$ with nonnegative Hessian by piecewise linear splines $s\in S(\triangle_N)$, generated by given triangulations $\triangle_N$ with $N$ elements. We find the exact asymptotic behavior of optimal (over triangulations $\triangle_N$ and splines $s\in S(\triangle_N)$ error of such approximation as $N\to \infty$.