it is well-known that if then the infinite Blaschke product (see Blaschke product - Wikipedia, the free encyclopedia) converges uniformly on compacts of the unit disk to a bounded holomorphic function f (so f is in the Hardy space , and then as well of course, see Hardy space - Wikipedia, the free encyclopedia)
I want to know if this product converges to f in sense (other way of putting this, -convergence of the boundary values of those functions)
in general, convergence is stronger than convergence on compacts: e.g. for any g in , define . Then uniformly on compacts of the unit disk, but doesn't go to zero. However I'm thinking that with the condition , this problem shouldn't occur.