On the convergence of multivariable Lagrange interpolants (Q1824733)

The author considers a triangular array of nodes in the compact set \(X\subseteq {\mathbb{C}}^ n\) and, for the analytic function f in a neighborhood of X, the corresponding Lagrange interpolant \(L_ d(f)\) of degree d at the given nodes. The main object of the paper is to obtain conditions of uniformly convergence of the sequence \(\{L_ d\}_{d\geq 1}\) to f on X. For example, if X is locally regular the author constructs a function \(\phi\) satisfying the Monge-Ampère equation on \({\mathbb{C}}^ n\setminus X\) such that if f is analytic on \(\{\) \(\phi\leq R\}\), \(R>1\), then \[ \| L_ d(f)-f\|_ X\leq B \exp (-d\cdot \log R), \] with convenable B. Numerous examples and relations between results are also presented.