Analytic Bounded Point Evaluations for Spaces of Rational Functions

Abstract

For a compact subset of the complex plane K and a regular Borel measure μ supported on K, R p ( K, μ) denotes the closure in L p (μ) of thè rational functions with poles off K. This paper examines the existence of analytic bounded point evaluations on R p ( K, μ) and extends the work of Thomson on the existence of analytic bounded point evaluations for the closure in L p (μ) of the analytic polynomials. Provided R p ( K, μ) is "pure," it is shown that the closure of the set of analytic bounded point evaluations for R p ( K, μ) equals the closure of the interior of the spectrum of the operator multiplication by z on R p ( K, μ). In fact this is derived as a consequence of Thomson′s result for polynomials. The bulk of the paper is devoted to certain interpolation problems for R p ( K, μ), thus producing some information on the structure of R p ( K, μ).