Exact Asymptotics of the Optimal Lp-error of Asymmetric Linear Spline Approximation
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 $N\to \infty$.
Jaen Journal on Approximation
Babenko, Vladyslav; Babenko, Yuliya; Parfinovych, Nataliya; and Skorokhodov, Dmytro, "Exact Asymptotics of the Optimal Lp-error of Asymmetric Linear Spline Approximation" (2014). Faculty Publications. 4029.