Blaschke Products with Prescribed Radial Limits

Abstract

If/ is an arbitrary function defined in the open unit disc D, e is a point of the unit circle and y is the radius from 0 to e, the radial cluster set of /a t e is the set of points a e C such that there exists a sequence {zn} in y with limn_oozn = e , such that \imn_00 f{zn) = a. C. Belna, P. Colwell and G. Piranian [1] have proved the following more general result. Let E = {e} be a countable subset of the unit circle and let {Km} be a sequence of nonempty, closed and connected subsets of the closed unit disc. Then, there exists a Blaschke product such that its radial cluster set at e is Km,m=\,2,.... Our aim is to extend these results to more general sets E, in the case of dealing with radial limits. By the F. and M. Riesz theorem, a bounded analytic function in the unit disc is determined by its radial limits at a set of positive measure of the circle. So, if we try to interpolate general functions by radial limits of Blaschke products, it is natural to restrict ourselves to subsets E of the unit circle of zero Lebesgue measure. A set is called of type Fa if it is a countable union of closed sets, and it is called of type Gd if it is a countable intersection of open sets. Observe that a closed subset of the unit circle is of type Fa and Gs. The closure of a set E will be denoted by E. Our result is the following.