Bounded Point Evaluations For Certain Polynomial And Rational Modules

TL;DR

This paper investigates bounded point evaluations for polynomial and rational modules over compact subsets of the complex plane, establishing conditions under which such evaluations exist and characterizing the relationship between various function algebras.

Contribution

It proves the existence of bounded point evaluations for certain modules and characterizes when the algebra of analytic functions is contained in these modules, especially for functions like  z,  z^2, ...,  z^N.

Findings

Existence of bounded point evaluations under certain conditions.

Characterization of when A(K) is contained in HP( z,  z^2, ...,  z^N, K).

Counterexamples showing limitations of the main results.

Abstract

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