Dense subset of holomorphic $L^2$ functions

Question

I've been having trouble with the following problem for a few weeks now for which it seems there should be an elementary proof.

Let $U \subset \mathbb{C}^n$ be open, connected and bounded with smooth boundary. Let $A^2(U) = \mathcal{O}(U) \cap L^2(U)$, i.e., the subspace of $L^2(U)$ consisting of holomorphic functions.

Claim: $A^2(U) \cap C(\overline{U})$ is dense in $A^2(U)$

Note: the norm we want to show this subset is dense with respect to is the $L^2$ norm.

The problem comes from Jiri Lebl's text ``Tasty Bits of Several Complex Variables", exercise 5.2.6, linked here: https://www.jirka.org/scv/scv.pdf

I also managed to track down a paper by David Catlin titled ``Boundary Behavior of Holomorphic Functions on Pseudoconvex Domains" which proves a more general result to do with holomorphic Sobolev Spaces, albeit, under a greater amount of assumptions and in much less elementary manner.

I've tried a whole bunch of different ideas like patching together Taylor series from individual discs using a partition of unity, to convoluting with smooth functions and then trying to show the resulting function is holomorphic.

The Taylor series idea comes from this argument: $L^2$ convergence of Taylor series of a holomorphic function but I ran into trouble with the convergence of Taylor series.

Something that might be helpful is that the polynomials are all in $A^2(U) \cap C(\overline{U})$.

I'm also interested in the result for values other than $p=2$.

Answer 1

I am posting here because the comments are already quite cluttered.

WARNING: This is not really an answer, because the holomorphic polynomials are not always dense in $C(\overline U)$; see Jiri Lebl's comment.

If you can prove that the set of all polynomials $P$ is dense in $C(\overline U)$ with respect to the $L^2(U)$ norm, as you claim in the comments, then you are done. Use sequences. You need to show that, for each $f\in A^2(U)$, there exists a sequence $p_n\in P$ such that $$\tag{1}\lVert p_n-f\rVert_2\to 0.$$ Now, since $f\in A^2\subset L^2$, there exists a sequence $g_n\in C(\overline U)$ such that $$\tag{2} \lVert g_n-f\rVert_2\to 0.$$ For each $n$, you can construct an approximation $p_{n, k}\in P$ to $g_n$; $$\tag{3} \lim_{k\to\infty} \lVert p_{n, k}-g_n\rVert_2=0.$$ Ok, now let $p_n:=p_{n,n}$.