Some Interpolation Problems Related To Blaschke Products

Blaschke product is of great significance to the study of zero points distribution of analytic functions on the unit disk,and it is also the direct source of many related examples.Because of its form is relatively easy to calculate,some scholars especially investigated the interpolation properties of Blaschke product in the process of studying Blaschke product.This article will present some of the interesting results in this area.In the first chapter,we briefly introduce the central issues concerned in this paper and the relevant works of other scholars,as well as some relevant symbols,definitions and basic results.In chapter 2,we mainly state some related properties of inner functions and Blaschke products,prepared for the following.In chapter 3,we mainly introduce Cantor,Phelps and G.T.Cargo’s works on some interpolation problems related to Blaschke products.In article[5],Cantor and Phelps discussed the existence of the interpolation problem for a finite number of points on the unit circle.G.T.Cargo gave an answer to the interpolation problem of the radial limit of a finite number of points on the unit circle in article [1].In chapter 4,we mainly introduce a generalization of the G.T.Cargo’s result given in paper [1].The generalization was proved by C.L.Belna,P.Colwell and G.Piranian in article [6].They showed that G.T.Cargo’s result presented in paper [1] can be extended not only to countably infinite sets but also to radial cluster sets.In paper [17] and [18],J.Earl introduced a more general interpolation problem for the results presented by Cantor and Phelps in paper [5].we review J.Earl’s this result in chapter 5.