On universal quotient Blaschke products

Abstract

In this paper, we introduce a new member to the universal families, called universal quotient Blaschke product, which is a formal quotient of two formal infinite Blaschke products. A formal infinite Blaschke product is of the form B ( z ) = ∏ k = 1 ∞ z − z k 1 − z k ¯ z , where { z k } k = 1 ∞ is a sequence of points in the unit disk but may not satisfy the Blaschke condition: ∑ k = 1 ∞ ( 1 − | z k | ) < ∞ . A partial quotient of a universal quotient Blaschke product is the quotient of two finite Blaschke products. We show that the set of partial quotients of a universal quotient Blaschke product is dense in the set of continuous self-mappings on the unit circle in the complex plane. Meanwhile, subsequences of the partial quotients of a universal quotient Blaschke product can be used to approximate any holomorphic functions bounded by one on the unit disk. Moreover, we prove that the set of universal quotient Blaschke products is huge in the sense of Baire category.