``Near-best'' polynomial approximation of harmonic functions on compact sets in \(\mathbb{C}\) (Q2054273)

It is established if \(u\) is a continuous function on the compact set \(K\) and harmonic on \(\mathring{K}\) (the interior of \(K\)) then there exists a sequence of polynomial approximants that give almost optimal rate of approximation on \(K\), and converges faster on \(\mathring{K}\) than on the whole \(K\). It is also shown that the geometric convergence inside \(K\) is possible for sets whose boundary is an analytically bounded Jordan domain.