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 $A(K) = HP (\phi ,K).$