Weierstrass factorization with $L^2$ estimates? Let $\Omega$ be a bounded domain in $\mathbb{C}$.  Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$.  Can I find an $L^2$ holomorphic function which vanishes on $X$?  Can I solve the problem in weighted $L^2$ spaces?
If there are counterexamples, are precise conditions on the set $X$ known to ensure the existence of an $L^2$ solution?
I have been learning about Hormander's approach to the $\bar{\partial}$-problem, and this seems like a natural question to ask from that perspective, but I have not been able to find any work done on this.
 A: You have to specify what you mean by $L^2$. Is this $L^2$ with respect to Lebesgue measure (area)
in $D$? Whatever you mean by $L^2$, the answer is "no". The reason is Jensen's formula. It says that
a function which has too many zeros must grow fast.
If you want to solve it in weighted $L^2$ space, then your weight must be related to the growth
rate of the set $X$. If instead you want to fix the weight in $L^2$, the conditions on $X$ will
come from the Jensen formula. If you are interested in $L^2$ without weight, look in the
books about Bergman space. There you can find the exact conditions on $X$.
A: The right place to start is the seminal work of Seip, 
Kristian Seip, Beurling type density theorems in the unit disk, Invent. math.
1993, Vol 113,  1, pp 21-39 (look at the last sections). You will see that much is known in the case of the disc, but I fairly doubt that a complete characterization is known for arbitrary domains (especially in the weighted case).
Also a good reference would be Ohsawa, T. 
On the extension of L2 holomorphic functions. V. Effects of generalization. Nagoya Math. J. 161 (2001), 1–21.  (again look just at the last chapters)
