In my complex analysis class we've shown that if $\Omega \subset \Bbb C^n$ is compact and if $H^2(\Omega)$ is the set of square integrable holomorphic functions on $\Omega$, then the Bergman kernel is defined as $$K(z,w)=\sum_{j=1}^{\infty}\phi_j(z)\overline{\phi_j(w)}$$ where $\{\phi_j\}_{j=1}^{\infty}$ is an orthonormal basis for $H^2(\Omega)$. However, my professor claimed without proof that the Bergman kernel is uniformly convergent. I'm unable to prove this and every place in the literature I've looked at just states it as a fact, so even a simple proof of this property would be much appreciated!
Uniform Convergence of Bergman Kernel
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complex-analysis
several-complex-variables
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0Here's a proof: http://books.google.co.uk/books?id=Sdv1uI5CCo8C&lpg=PA189&ots=WHkhHKXPek&pg=PA189#v=onepage&q&f=false (page 189 of Analysis and Geometry on Complex Homogeneous Domains by Jacques Faraut). – 2012-04-11
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0This just came up in a different question: http://math.stackexchange.com/questions/212079/uniform-convergence-of-the-bergman-kernels-orthonormal-basis-representation-on – 2012-10-18