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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Edwin Wolf
Journal: Trans. Amer. Math. Soc. 238 (1978), 71-88
MSC: Primary 46E99
DOI: https://doi.org/10.1090/S0002-9947-1978-0470679-X
MathSciNet review: 0470679
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0470679-X
Keywords: Rational functions, compact set, $ {L^p}$-spaces, bounded point evaluation, representing function, admissible function, full area density, $ {\Gamma _q}$-capacity
Article copyright: © Copyright 1978 American Mathematical Society