On The Boundary Behavior Of Blaschke Products In The Unit Disk

As a class of bounded analytic functions on open unit disc.The Blaschke product has been widely concerned because it plays an important role in the study of the zero distribution of analytical functions.This article mainly describes the behavior of Blaschke products near the boundary,especially,the boundary behavior of Blaschke products under Frostman conditions and give some results about boundary behavior of Cowen-Douglas operator on their spectra.In the first part of the introduction,we will briefly give some related concepts of Blaschke products and important theorems,and provide theoretical basis for the fol-lowing proofs.In the second chapter,we mainly state the behavior of Blaschke products near the boundary,especially,the existence of the radial limit of Blaschke products in the unit circle T,which makes the necessary preparations for the rest of the article.The third chapter,we mainly introduce the boundary behavior of Blaschke products under Frostman conditions.In particular,the work of P.Colwell is reviewed:under Frostman conditions,the accumulation point set of zero points of Blaschke products must be closed and nowhere dense on the unit circle T,then we will introduce the work of Linden:by selecting the argument,let Blaschke products diverge everywhere on the unit disc.Their work is closely related with Frostman.In the fourth chapter,we mainly describe the recent work on Frostman Blaschke products,on the one hand,introduces the boundary spectra of a uniform Frostman Blaschke product,and on the other hand,we will introduce the composition of the Frostman Blaschke product.J.R.Akeroyd and P.Gorkin had given an example,for the study of the composition of Frostman Blaschke products,which has important significance.At the end of this paper,we introduce the recent work of our group and describe the boundary behavior of Cowen-Douglas operator on their spectra.