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

Abstract

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

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• Optimal approximation order of piecewise constants on convex partitions