A construction of Hilbert spaces of analytic functions

Abstract:

A simple technique is presented for constructing a measure $\nu$ from a given measure so that ${R^2}(K,\nu )$ has certain properties. Here, ${R^2}(K,\nu )$ is the closure in ${L^2}(\nu )$ of the rational functions with poles off K, a compact set containing the support of $\nu$. A typical example shows that there exists a measure $\nu$ mutually absolutely continuous with area measure on the unit disk such that $\sqrt z$ (principal branch) is an element of ${R^2}(K,\nu )$, while each point of the open disk except on the negative real axis is an analytic bounded point evaluation for ${R^2}(K,\nu )$.