Bounded point evaluations for certain polynomial and rational modules.

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) ⊂ C (K) be the algebra of functions that are analytic in the interior of K. For 1 ≤ t < ∞ , let P t (1 , ϕ 1 ,... , ϕ N , K) be the closure of P (K) + P (K) ϕ 1 +... + P (K) ϕ N in L t (d A | K) , where d A | K is the area measure restricted to K and ϕ 1 ,... , ϕ N ∈ L t (d A | K). Let H P (ϕ 1 ,... , ϕ N , K) be the closure of P (K) ϕ 1 +... + P (K) ϕ N + R (K) in C (K) , where ϕ 1 ,... , ϕ N ∈ C (K). In this paper, we prove if R (K) ≠ C (K) , then there exists an analytic bounded point evaluation for both P t (1 , ϕ 1 ,... , ϕ N , K) and H P (ϕ 1 ,... , ϕ N , K) for certain smooth functions ϕ 1 ,... , ϕ N , in particular, for z ¯ , z ¯ 2 ,... , z ¯ N. We show that A (K) ⊂ H P (z ¯ , z ¯ 2 ,... , z ¯ N , K) if and only if R (K) = A (K). In particular, C (K) ≠ H P (z ¯ , z ¯ 2 ,... , z ¯ N , K) unless R (K) = C (K). We also give an example of K showing the results are not valid if we replace z ¯ n by certain ϕ n , that is, there exist K and a function ϕ ∈ A (K) such that R (K) ≠ A (K) , but A (K) = H P (ϕ , K). [ABSTRACT FROM AUTHOR]