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

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

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

any thoughts?