Some remarks on the space R2(E)

Abstract

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≤p1, 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.

Keywords

• rational functions
• compact set
• Lp-spaces
• bonded point evaluation
• weak bounded point evaluation
• Bessel capacity.