Bounded point evaluations and smoothness properties of functions in 𝑅^{𝑝}(𝑋)

Abstract

Let X be a compact subset of the complex plane C. We denote by R 0 ( X ) {R_0}(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 p \geqslant 1 , let L p ( X ) = L p ( X , d m ) {L^p}(X) = {L^p}(X,dm) . The closure of R 0 ( X ) {R_0}(X) in L p ( X ) {L^p}(X) will be denoted by R p ( X ) {R^p}(X) . Whenever p and q both appear, we assume that 1 / p + 1 / q = 1 1/p + 1/q = 1 . If x is a point in X which admits a bounded point evaluation on R p ( X ) {R^p}(X) , then the map which sends f to f ( x ) f(x) for all f ∈ R 0 ( X ) f \in {R_0}(X) extends to a continuous linear functional on R p ( X ) {R^p}(X) . The value of this linear functional at any f ∈ R p ( X ) f \in {R^p}(X) is denoted by f ( x ) f(x) . We examine the smoothness properties of functions in R p ( X ) {R^p}(X) at those points which admit bounded point evaluations. For p > 2 p > 2 we prove in Part I a theorem that generalizes the “approximate Taylor theorem” that James Wang proved for R ( X ) R(X) . In Part II we generalize a theorem of Hedberg about the convergence of a certain capacity series at a point which admits a bounded point evaluation. Using this result, we study the density of the set X at such a point.