Some remarks on the space R2(E)

Let E be a compact subset of the complex plane. We denote by R(E) the algebra consisting of the rational functions with poles off E. The closure of R(E) in Lp(E), 1≤p<∞, is denoted by Rp(E). In this paper we consider the case p=2. In section 2 we introduce the notion of weak bounded point evaluation of order β and identify the existence of a weak bounded point evaluation of order β, β>1, as a necessary and sufficient condition for R2(E)≠L2(E). We also construct a compact set E such that R2(E) has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions in R2(E) at those points which admit bounded point evaluations