Kergin interpolants of holomorphic functions (Q1381511)

Let \(D\subset\mathbb{C}^n\) be a \(\mathbb{C}\)-convex domain, that is, its intersection with each complex line \(\ell\) is a simply connected domain in \(\ell\). Let \(\{a_{nm}\}\), \(n= 1,2,\dots\), \(m= 1,2,\dots, n\) be an array of points in a compact set \(K\subset D\). Then, it is shown that for a function \(f\) holomorphic in \(D\) there exists a polynomial \(P_n\) in \(\mathbb{C}^n\) of degree \(\leq n\) such that \(P(a_{nm}\}= f(a_{nm})\), \(m= 1,2,\dots, n\). An integral formula for the error \(f(z)- P_n(z)\) is proved. It is a generalization of the one-dimensional formula of Hermite. This formula is used to prove the main result of the paper: a theorem giving conditions which imply the uniform convergence of \(P_n(f)\) to \(f\) on \(K\). These conditions are given in terms of potential theory.