Some remarks on the spaceR2(E)

LetEbe a compact subset of the complex plane. We denote byR(E)the algebra consisting of the rational functions with poles offE. The closure ofR(E)inLp(E),1≤p<∞, is denoted byRp(E). In this paper we consider the casep=2. In section 2 we introduce the notion of weak bounded point evaluation of orderβand identify the existence of a weak bounded point evaluation of orderβ,β>1, as a necessary and sufficient condition forR2(E)≠L2(E). We also construct a compact setEsuch thatR2(E)has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions inR2(E)at those points which admit bounded point evaluations.

Functions in the spaceR2(E)at boundary points of the interior

LetEbe a compact subset of the complex planeℂ. We denote byR(E)the algebra consisting of (the restrictions toEof) rational functions with poles offE. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(E)be the closure ofR(E)inLp(E,dm).In this paper we consider the casep=2. Letx ϵ ∂Ebe a bounded point evaluation forR2(E). Suppose there is aC>0such thatxis a limit point of the sets={y|y ϵ Int E,Dist(y,∂E)≥C|y−x|}. For thosey ϵ Ssufficiently nearxwe prove statements about|f(y)−f(x)|for allf ϵ R(E).

Smoothness properties of functions inR2(x)at certain boundary points

LetXbe a compact subset of the complex planeℂ. We denote byR0(X)the algebra consisting of the (restrictions toXof) rational functions with poles offX. Letmdenote2-dimensional Lebesgue measure. Forp≥1, letRp(X)be the closure ofR0(X)inLp(X,dm).In this paper, we consider the casep=2. Letxϵ∂Xbe both a bounded point evaluation forR2(X)and the vertex of a sector contained inIntX. LetLbe a line which passes throughxand bisects the sector. For thoseyϵL∩Xthat are sufficiently nearxwe prove statements about|f(y)−f(x)|for allfϵR2(X).