it is well-known that if \sum_n 1-|z_n|<\infty then the infinite Blaschke product (see Blaschke product - Wikipedia, the free encyclopedia) \prod_n \frac{|z_n|}{z_n}\frac{z-z_n}{1-\overline{z_n}z} converges uniformly on compacts of the unit disk to a bounded holomorphic function f (so f is in the Hardy space H^\infty, and then H^2 as well of course, see Hardy space - Wikipedia, the free encyclopedia)

I want to know if this product converges to f in H^2 sense (other way of putting this, L^2-convergence of the boundary values of those functions)

in general, H^2 convergence is stronger than convergence on compacts: e.g. for any g in H^2, define g_n(z)=z^n g(z). Then g_n(z)\to 0 uniformly on compacts of the unit disk, but ||g_n||_2=||g||_2 doesn't go to zero. However I'm thinking that with the condition \sum_n 1-|z_n|<\infty, this problem shouldn't occur.

any thoughts?