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Good evening,

I have a question on the approximation of holomorphic functions on a space of cartesian product type.

Question: Let $U,V$ be domains in $\mathbb{C}^n$ and $f\in \mathcal{O}(U\times V)$ a holomorphic function on $U\times V.$ Do we always have the following : $f$ can be approximated by holomorphic functions on $U\times V$ of the form $g(z,w) = \sum_{i=1}^N h_i(z)k_i(w)$ where $h_i\in \mathcal{O}(U)$ and $k_i\in\mathcal{O}(V)$ ? (N is arbitrary)

If it is not possible, can this be true if we put some conditions on $U$ and $V$? So what are the conditions?

Any help is appreciated. Thanks in advance.

Duc Anh

flag	asked Jun 13 at 18:55 Đức Anh 485●1●3●8

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I would suspect that the answer is no. It may be true if U and V are domains of holomorphy. I will think about it and post an answer shortly. – Steven Gubkin Jun 13 at 19:27

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This is just a comment since I have not checked the details, but one special case where this might work is when $U$ and $V$ are open polydiscs, since then $U\times V$ is also an open polydisc and every holomorphic function $f$ on an open polydisc admits a power series expansion about the centre; truncating that series at some $N$ will give functions $g_N$ of the form you describe, and then my guess is that the functions $g_N$ converge locally uniformly to $f$. – Yemon Choi Jun 14 at 3:46

Thank you. By the way, I think if $U, V$ are polynomially convex, we can get this kind of approximation. But it is very special :D, because the approximating functions are polynomials. – Đức Anh Jun 14 at 7:27

Đức Anh · Answer 1 · 2012-06-15 16:32:14Z UTC

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This is the answer : the above statement is true without any conditions on $U$ and $V.$ It is the theorem 1.7.7 in the book of Narasimhan, Analysis on Real and Complex Manifolds. One of my professors has pointed it out for me.

answered Jun 15 at 16:32

Đức Anh
485●1●3●8

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You can accept your own answer, to stop the question being listed as "open" – Yemon Choi Jun 15 at 18:18

You're right, but I don't want to do it, because i will feel uncomfortable (don't ask me why). Anyway, we had an answer, it is sufficient. – Đức Anh Jun 15 at 18:39

Đức Anh · Answer 2 · 2012-06-15 10:35:56Z UTC

This is not an answer, but I hope someone will read and explain a little. I think the answer is the theorem 4.2.4, page 107, Eschmeier, Putinar, Spectral Decompostions and Analytic Sheaves. From the theorem, my above statement will be true if $U$ and $V$ are Stein spaces. To understand the statement of the theorem and also its proof, we have to know coherent sheaves, tensor product of two locally convex spaces, etc. So it is very far from my knowledge.

on the approximation by holomorphic functions

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