Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$begin{gathered} E (H_Phi ^omega ; P (Delta _n )) = sup || f - l_{p(Delta _n )} (f)||, hfill f in H_Phi ^omega hfill end{gathered} $$ and also give an asymptotic estimate of the quantities $$begin{gathered} E_n (H_Phi ^omega ) = infinf E (H_Phi ^omega ; P (Delta _n )). hfill Delta _n P(Delta _n ) hfill end{gathered} $$