One of my homework problems this week is to "characterize all holomorphic functions in $L^2(\Bbb C^n)$". I'm sorry for not being able to provide much work on my progress, but that is because I really don't know where to begin. Any help would be greatly appreciated!
Characterizing holomorphic functions in $L^2(\Bbb C^n)$
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complex-analysis
several-complex-variables
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3Write down some holomorphic functions (say when $n = 1$ for simplicity) and check whether they lie in $L^2$. Repeat. Make a conjecture. See if you can prove it. – 2012-04-10
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1Well if a function is in $L^2(\mathbb{C}^n)$, what can you say about it as $|z| \rightarrow \infty$? Then ask yourself what holomorphic functions can satisfy such a condition. – 2012-04-10
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1@MichaelJoyce: I'm not sure what you're getting at but I don't think there is a simple condition satisfied by $L^2$ functions at infinity. – 2012-04-10
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1The mean value problem for holomorphic functions will probably be useful. – 2012-04-10
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0@Eric: Oops, I initially thought $L^2$ + continuous would imply bounded, but that's not the case. – 2012-04-10
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0A variation of Liouville's theorem – 2012-05-14