author_facet Wolf, Edwin
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Transactions of the American Mathematical Society
Bounded Point Evaluations and Smoothness Properties of Functions in $R^p(X)$
author_sort wolf, edwin
spelling Wolf, Edwin 0002-9947 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 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 $p &gt; 2$ 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
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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 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 $p &gt; 2$ 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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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 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 $p &gt; 2$ 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 0002-9947 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 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 $p &gt; 2$ 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