Vladislav Babenko, Yuliya Babenko, Nataliya Parfinovych et al.
In this paper, we show that the $L_p$-error of asymmetric linear approximation of the quadratic function $Q({\mathbf x})=\sum_{j=1}^{d}x_j^2$ on simplices in $\RR^d$ of fixed volume is minimized on regular simplices.
Vladislav Babenko, Yuliya Babenko, Dmytro Skorokhodov
In this paper we provide the exact asymptotics of the optimal weighted $L_p$-error, $0<p< \infty$, of linear spline interpolation of $C^2$ functions with positive Hessian. The full description of the behavior of the optimal error leads to the algorithm for construction of an asymptotically optimal sequence of triangulations. In addition, we compute the minimum of the $L_p$-error of linear interpolation of the function $x^2+y^2$ over all triangles of unit area for all $0<p<\infty$. This provides the exact constant in the asymptotics of the optimal error.