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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 

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In:  Transactions of the American Mathematical Society, 238, 1978, p. 7188 
Type of Resource:  EArticle 
Language:  English 
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The American Mathematical Society

author_facet 
Wolf, Edwin Wolf, Edwin 

author 
Wolf, Edwin 
spellingShingle 
Wolf, Edwin Transactions of the American Mathematical Society Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
author_sort 
wolf, edwin 
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Wolf, Edwin 00029947 The American Mathematical Society https://www.jstor.org/stable/1997798 <p>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 2dimensional 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.</p> Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Transactions of the American Mathematical Society 
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title 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_unstemmed 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_full 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_fullStr 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_full_unstemmed 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_short 
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_sort 
bounded point evaluations and smoothness properties of functions in $r^p(x)$ 
url 
https://www.jstor.org/stable/1997798 
publishDate 
1978 
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7188 
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<p>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 2dimensional 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.</p> 
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author  Wolf, Edwin 
author_facet  Wolf, Edwin, Wolf, Edwin 
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container_title  Transactions of the American Mathematical Society 
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description  <p>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 2dimensional 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.</p> 
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spelling  Wolf, Edwin 00029947 The American Mathematical Society https://www.jstor.org/stable/1997798 <p>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 2dimensional 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.</p> Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ Transactions of the American Mathematical Society 
spellingShingle  Wolf, Edwin, Transactions of the American Mathematical Society, Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_full  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_fullStr  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_full_unstemmed  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_short  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
title_sort  bounded point evaluations and smoothness properties of functions in $r^p(x)$ 
title_unstemmed  Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$ 
url  https://www.jstor.org/stable/1997798 