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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

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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 '12 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 '12 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 '12 at 7:27
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2 Answers

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.

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You can accept your own answer, to stop the question being listed as "open" –  Yemon Choi Jun 15 '12 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 '12 at 18:39
    
Do you have a way to construct such an approximation? –  Mikaël Mayer 7 hours ago
    
hi, at this time, i don't know, because i asked this question too long time ago, so i forgot lots of things. But did you take a look at the proof in the book of Narasimhan? I think there will have a way. –  Đức Anh 3 hours ago
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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.

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