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# Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$

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Journal Title: Transactions of the American Mathematical Society E-Article English
openURL url_ver=Z39.88-2004&ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fvufind.svn.sourceforge.net%3Agenerator&rft.title=Bounded+Point+Evaluations+and+Smoothness+Properties+of+Functions+in+%24R%5Ep%28X%29%24&rft.date=1978&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.creator=Wolf%2C+Edwin&rft.pub=The+American+Mathematical+Society&rft.format%5B0%5D=ElectronicArticle&rft.language=English 1759245603823419400 Electronic Resources Wolf, Edwin Wolf, Edwin, Wolf, Edwin wolf, edwin Electronic Resources sid-55-col-jstoras1, sid-55-col-jstormaths 71 Transactions of the American Mathematical Society 238

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.

Online, Free ElectronicArticle Article, E-Article Article, E-Article Article, E-Article Article, E-Article Article, E-Article Article, E-Article Article, E-Article E-Article Buch Article, E-Article Article Article, E-Article Article, E-Article Article, E-Article not assigned not assigned ai-55-aHR0cHM6Ly93d3cuanN0b3Iub3JnL3N0YWJsZS8xOTk3Nzk4 The American Mathematical Society, 1978 The American Mathematical Society, 1978 DE-14, DE-15, DE-Ch1, DE-D13 0002-9947 0002-9947 English 2023-03-02T08:55:48.167Z wolf1978boundedpointevaluationsandsmoothnesspropertiesoffunctionsinrpx JSTOR Arts & Sciences I Archive, JSTOR Mathematics & Statistics 71-88 1978 1978 The American Mathematical Society ai 18,660866 Transactions of the American Mathematical Society 55 Wolf, Edwin 0002-9947 The American Mathematical Society https://www.jstor.org/stable/1997798

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.

Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Transactions of the American Mathematical Society Wolf, Edwin, Transactions of the American Mathematical Society, Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ bounded point evaluations and smoothness properties of functions in $r^p(x)$ https://www.jstor.org/stable/1997798