Asymptotic polynomial approximation in the Bloch space

Abstract: We investigate asymptotic polynomial approximation for a class of weighted Bloch functions in the unit disc. Our main result is a structural theorem on asymptotic polynomial approximation in the unit disc, in the flavor of the classical Plessner Theorem on asymptotic values of meromorphic functions. This provides the appropriate set up for studying metric and geometric properties of sets E on the unit circle for which the following simultaneous approximation phenomenon occurs: there exists analytic polynomials which converge uniformly to zero on E and to a non-zero function in the weighted Bloch norm. We offer a characterization completely within the realm of real-analysis, establish a connection to removable sets for analytic Sobolev functions in the complex plane, and provide several necessary conditions in terms of entropy, Hausdorff content and condenser capacity. Furthermore, we demonstrate two principal applications of our developments, which go in different directions. First, we shall deduce a rather subtle consequence in the theme of smooth approximation in de Branges-Rovnyak spaces. Secondly, we answer some questions that were raised almost a decade ago in the theory of Universal Taylor series.