paper

Bounded Point Evaluations For Certain Polynomial And Rational Modules

arXiv:1711.03715

Abstract

Let $K$ be a compact subset of the complex plane $C .$ 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 $1 \leq t < \infty,$ let $P^{t} (1, I ¨ _{1}, ..., I ¨ _{N}, K)$ be the closure of $P (K) + P (K) I ¨ _{1} + ... + P (K) I ¨ _{N}$ in $L^{t} (d A ∣_{K}),$ where $d A ∣_{K}$ is the area measure restricted to $K$ and $I ¨ _{1}, ..., I ¨ _{N} \in L^{t} (d A ∣_{K}) .$ Let $H P (I ¨ _{1}, ..., I ¨ _{N}, K)$ be the closure of $P (K) I ¨ _{1} + ... + P (K) I ¨ _{N} + R (K)$ in $C (K),$ where $I ¨ _{1}, ..., I ¨ _{N} \in C (K) .$ In this paper, we prove if $R (K) \neq = C (K),$ then there exists an analytic bounded point evaluation for both $P^{t} (1, I ¨ _{1}, ..., I ¨ _{N}, K)$ and $H P (I ¨ _{1}, ..., I ¨ _{N}, K)$ for certain smooth functions $I ¨ _{1}, ..., I ¨ _{N},$ in particular, for $z ˉ, z ˉ^{2}, ..., z ˉ^{N} .$ We show that $A (K) \subset H P (z ˉ, z ˉ^{2}, ..., z ˉ^{N}, K)$ if and only if $R (K) = A (K) .$ In particular, $C (K) \neq = H P (z ˉ, z ˉ^{2}, ..., z ˉ^{N}, K)$ unless $R (K) = C (K) .$ We also give an example of $K$ showing the results are not valid if we replace $z ˉ^{n}$ by certain $I ¨ _{n},$ that is, there exist $K$ and a function $I ¨ \in A (K)$ such that $R (K) \neq = A (K),$ but $A (K) = H P (I ¨ , K) .$