Boundary Behavior of Universal Taylor Series and Their Derivatives

Abstract

For a given first category subset E of the unit circle and any given holomorphic function g on the open unit disk, we construct a universal Taylor series f on the open unit disk, such that, for every n = 0,1,2,..., f(n) is close to g(n) on a set of radii having endpoints in E. Therefore, there is a universal Taylor series f, such that f and all its derivatives have radial limits on all radii with endpoints in E. On the other hand, we prove that if f is a universal Taylor series on the open unit disk, then there exists a residual set G of the unit circle, such that for every strictly positive integer n, the derivative f(n) is unbounded on all radii with endpoints in the set G.