Transfinite interpolation on the medians of a triangle and best \(L^1\)-approximation (Q1779990)

It is known that, on special domains \(I\), best \({L^1(I)}\)-approximations from specific function spaces can be constructed through interpolation on canonical interpolation sets, subject that the function to be approximated is from a certain convexity cone. The most famous result in this direction is the Theorem of A.A.~Markov on \({L^1[-1,+1]}\)-approximation (Theorem 10.10 in [\textit{R. A. DeVore} and \textit{G. G. Lorentz}, Constructive approximation (Berlin: Springer-Verlag) (1993; Zbl 0797.41016)]). Multivariate versions of the Markov theorem are known only for very specific domains (cubes, balls, or strips). The authors obtain the result of this type in the following settings. The basic domain is a triangle \(\triangle\) in \(R^2\). The interpolation set is the union of the three medians \({{\mathcal M}={\mathcal M_1}\cup{\mathcal M_2}\cup{\mathcal M_3}}\). A function \({F\in C^3(\triangle)}\) to be approximated satisfy \({{\mathcal D}F\geq 0}\) on \(\triangle\), where \({\mathcal D}\) is defined in Section~3. Approximants are taken from the set \({B(\triangle)}\) of functions of the form \[ H(\lambda_1,\lambda_2,\lambda_3)=h_1(\lambda_1)+h_2(\lambda_2)+h_3(\lambda_3), \] where \({\lambda_1,\lambda_2,\lambda_3}\) are barycentric coordinates. It is proved that there is a unique \(C^3\)-smooth function \(H_F\) such that \({H_F| {\mathcal M}=F| {\mathcal M}}\) and this function is the unique best \(L^1\)-approximant of \(F\).