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 ${R_0}(X)$ the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For $p \geqslant 1$, let ${L^p}(X) = {L^p}(X,dm)$. The closure of ${R_0}(X)$ in ${L^p}(X)$ will be denoted by ${R^p}(X)$. Whenever p and q both appear, we assume that $1/p + 1/q = 1$. If x is a point in X which admits a bounded point evaluation on ${R^p}(X)$, then the map which sends f to $f(x)$ for all $f \in {R_0}(X)$ extends to a continuous linear functional on ${R^p}(X)$. The value of this linear functional at any $f \in {R^p}(X)$ is denoted by $f(x)$. We examine the smoothness properties of functions in ${R^p}(X)$ at those points which admit bounded point evaluations. For $p > 2$ we prove in Part I a theorem that generalizes the “approximate Taylor theorem” that James Wang proved for $R(X)$. 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.