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.



Weierstrass factorization with $L^2$ estimates?
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)


