Analytic bounded point evaluations for spaces of rational functions (Q1316364)

For a compact subset of the complex plane \(K\) and a regular Borel measure \(\mu\) supported on \(K\), \(R^ p(K,\mu)\) denotes the closure in \(L^ p(\mu)\) of the rational functions with poles off \(K\). In this paper the existence of analytic bounded point evaluations on \(R^ p(K,\mu)\) is examined and the work of Thomson on the existence of analytic bounded point evaluations for the closure in \(L^ p(\mu)\) of the analytic polynomials is extended. Provided \(R^ p(K,\mu)\) is ``pure'', it is shown that the closure of the set of analytic bounded point evaluations for \(R^ p(K,\mu)\) equals the closure of the interior of the spectrum of the operator multiplication by \(z\) on \(R^ p(K,\mu)\). 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 \(R^ p(K,\mu)\), thus producing some information on the structure of \(R^ p(K,\mu)\).