Hilbert spaces of holomorphic functions

Question

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps some general properties. Thanks in advance!

Answer 1

For every continuous and $>0$ density $\mu(x) |dx|$ on a complex manifold $X$ the subspace of $L^2(X, \mu |dx|)$ consisting of homolorphic functions is a closed Hilbert subspace, so a Hilber space. The only problem is whether the space has enough functions, this will happen for instance for bounded open subsets of $\mathbb{C}^n$. See for example http://en.wikipedia.org/wiki/Bergman_space

Answer 2

The most classical space of holomorphic functions is the hardy space $H^{2}(D)$, where $D$ is the open unit disk $D$ in the complex plane. A holomorphic function on $D$ is in $H^{2}(D)$ iff $$ \|f\|^{2}_{H^{2}}=\sup_{0 < r < 1} \frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}\,d\theta < \infty. $$ Every $f \in H^{2}(D)$ automatically has an $L^{2}$ boundary limit $$ f_{1}(e^{i\theta}) = L^{2}-\lim_{r\uparrow 1}f(re^{i\theta}). $$ Furthermore, $f$ is the Cauchy integral of this boundary function. This space, and various generalizations to other domains have been studied extensively, and have proven useful in a wide wide of applications, including radar, MRI, and sending a missle to chase a target. I mention applications just to make sure Hardy is still turning over in his grave.

Reference: http://en.wikipedia.org/wiki/Hardy_space#Hardy_spaces_for_the_unit_disk