On Fatou's Theorem And An Extension Of Rudin

This treatise is an exposition of some fundamental results derived from Fatou's theorem concerning radial limits for bounded analytic functions in the open unit disc of the complex plane, presented in historical order. The main objective is to introduce Rudin's extension of Fatou's theorem, in which radial limits are broadened to the most general class of paths approaching the boundary.Chapter I contains some basic facts. In chapter II a proof of Fatou's theorem is given, together with the stronger Lindelof's theorem for angular limits of bounded analytic functions in the open unit disc. Chapter III deals with Blaschke products, introduced to illustrate the extent to which Fatou's theorem may be extended. Chapter IV presents the extension due to Rudin, and proves that this extension is the best possible. Finally, some open problems related to this area are stated.