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        general >> complex analysis >> Limit of a Blaschke product sequence
        
(Message started by: Sherlock on May 18^th, 2006, 6:43am)

Title: Limit of a Blaschke product sequence
Post by Sherlock on May 18^th, 2006, 6:43am

Hello everyone,

I'm puzzled by the following problem:

If a sequence {B^j} of Blaschke products
converges normally to a nonconstant holomorphic function B^0 on D, is B^0 a Blaschke product?

My hunch is that since every member of the sequence is a Blaschke product, the limit might be one as well---but maybe my thinking's too pedestrian.

Title: Re: Limit of a Blaschke product sequence
Post by Michael_Dagg on Jun 29^th, 2006, 6:05pm

Sherlock, have you discovered an answer to your
question?

Title: Re: Limit of a Blaschke product sequence
Post by Sherlock on Jul 3^rd, 2006, 3:11am

Actually I haven't. ??? Any ideas?

Title: Re: Limit of a Blaschke product sequence
Post by Michael_Dagg on Jul 6^th, 2006, 4:34pm

The functions (z - 1/n)/(z/n - 1) are Blaschke products
and converge uniformly in D to -z, which is a Blaschke product.

The result is different on D bar (equivalent to norm
convergence in L^{\infty} of the boundary).