Abstract

Let X be a compact subset of the complex plane ℂ. We denote by R0(X) the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For p≥1, let Rp(X) be the closure of R0(X) in Lp(X,dm).In this paper, we consider the case p=2. Let xϵ∂X be both a bounded point evaluation for R2(X) and the vertex of a sector contained in IntX. Let L be a line which passes through x and bisects the sector. For those yϵL∩X that are sufficiently near x we prove statements about |f(y)−f(x)| for all fϵR2(X).