Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounded Point Evaluations For Certain Polynomial And Rational Modules   Let $K$ be a compact subset of the complex plane $\mathbb C.$ Let $P(K)$ and
$R(K)$ be the closures in $C(K)$ of analytic polynomials and rational functions
with poles off $K,$ respectively. Let $A(K) \subset C(K)$ be the algebra of
functions that are analytic in the interior of $K$. For $1\le t <\infty,$ let
$P^t(1, \phi_1,...,\phi_N,K)$ be the closure of
$P(K)+P(K)\phi_1+...+P(K)\phi_N$ in $L^t(dA|_K),$ where $dA|_K$ is the area
measure restricted to $K$ and $\phi_1,...,\phi_N\in L^t(dA|_K).$ Let
$HP(\phi_1,...,\phi_N,K)$ be the closure of $P(K)\phi_1+...+P(K)\phi_N +R(K)$
in $C(K),$ where $\phi_1,...,\phi_N\in C(K).$ In this paper, we prove if
$R(K)\ne C(K),$ then there exists an analytic bounded point evaluation for both
$P^t(1, \phi_1,...,\phi_N,K)$ and $HP(\phi_1,...,\phi_N,K)$ for certain smooth
functions $\phi_1,...,\phi_N,$ in particular, for $\bar z,\bar z^2,...,\bar
z^N.$ We show that $A(K)\subset HP(\bar z,\bar z^2,...,\bar z^N,K)$ if and only
if $R(K) = A(K).$ In particular, $C(K) \ne HP(\bar z,\bar z^2,...,\bar z^N,K)$
unless $R(K) = C(K).$ We also give an example of $K$ showing the results are
not valid if we replace $\bar z^n$ by certain $\phi_n,$ that is, there exist
$K$ and a function $\phi\in A(K)$ such that $R(K) \ne A(K),$ but $A(K) = HP
(\phi ,K).$

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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat