on the approximation by holomorphic functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:57:01Z http://mathoverflow.net/feeds/question/99486 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99486/on-the-approximation-by-holomorphic-functions on the approximation by holomorphic functions Đức Anh 2012-06-13T18:55:55Z 2012-06-15T16:32:14Z <p>Good evening,</p> <p>I have a question on the approximation of holomorphic functions on a space of cartesian product type.</p> <p><strong>Question:</strong> 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)</p> <p>If it is not possible, can this be true if we put some conditions on $U$ and $V$? So what are the conditions?</p> <p>Any help is appreciated. Thanks in advance.</p> <p>Duc Anh</p> http://mathoverflow.net/questions/99486/on-the-approximation-by-holomorphic-functions/99690#99690 Answer by Đức Anh for on the approximation by holomorphic functions Đức Anh 2012-06-15T10:35:56Z 2012-06-15T10:35:56Z <p>This is not an answer, but I hope someone will read and explain a little. I think the answer is the <strong>theorem 4.2.4</strong>, page 107, <em>Eschmeier, Putinar, Spectral Decompostions and Analytic Sheaves.</em> 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.</p> http://mathoverflow.net/questions/99486/on-the-approximation-by-holomorphic-functions/99723#99723 Answer by Đức Anh for on the approximation by holomorphic functions Đức Anh 2012-06-15T16:32:14Z 2012-06-15T16:32:14Z <p>This is the answer : the above statement is <strong>true without any conditions</strong> 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. </p>