Transfinite interpolation on the medians of a triangle and best L1-approximation

Abstract

Let Δ be a triangle in $$ \mathbb{R}^{2} , $$ and let $$ \mathcal{M} $$ be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on $$ \mathcal{M}. $$ The interpolants are of type f(λ1)+g(λ2)+h(λ3), where (λ1,λ2,λ3) are the barycentric coordinates with respect to the vertices of Δ. Based on an error representation formula, we prove that the interpolant is the unique best L1-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C3(Δ).