Smoothness Properties of Functions in Rp(x)(x)

Let X be a compact subset of the complex plane C. We denote by R0(X) the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For/» > 1, let L"(X) = L"(X, dm). The closure of R0(X) in LP(X) will be denoted by Rr(X). Whenever p and q both appear, we assume that \/p + \/q = 1. If x is a point in X which admits a bounded point evaluation on RP(X), then the map which sends / to f(x) for all / £ R0(X) extends to a continuous linear functional on RP(X). The value of this linear functional at any/ e RP(X) is denoted by/(x). We examine the smoothness properties of functions in RP(X) at those points which admit bounded point evaluations. For p>2we 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. Part I. Smoothness properties of functions in Rp (X) Let A" be a compact subset of the complex plane C. We denote by R0(X) the algebra consisting of the (restrictions to A') of rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For p > 1, let Lp(X) = Lp(X, dm). The closure of R0(X) in L"(X) will be denoted by RP(X). Wheneverp and q both appear, we will assume that 1/p + l/q = 1. 1. Bounded point derivations. Definition (1.1). For x E X we say that x admits a bounded point derivation of order s on RP(X) if there exists a constant C such that |/{í)(jí:)| < CU/U, for aU/£*„(*). When x admits a bounded point derivation of order s on RP(X), the map f\-+f{s)(x)/s\ extends from R0(X) to a bounded linear functional on RP(X). Received by the editors July 13, 1976. AMS (MOS) subject classifications (1970). Primary 46-XX, 46EXX, 46E99.