on the approximation by holomorphic functions - MathOverflow

on the approximation by holomorphic functions

2012-06-13T18:55:55Z

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

Answer by Đức Anh for on the approximation by holomorphic functions

2012-06-15T10:35:56Z

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.

Answer by Đức Anh for on the approximation by holomorphic functions

2012-06-15T16:32:14Z

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.