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 ) .