Bergman spaces are basically the analytic functions that are absolutely integrable. They are denoted as $A^p(G) \subset L^p(G)$ where $G$ is a domain of $\mathbb{C}$

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$L^2$ convergence of Taylor series of a holomorphic function

I am reading Otto Forster's book "Lecture on Riemann surfaces" and on pages 109-110, he introduces the space $L^2(D,\mathcal{O})$ of holomorphic square-integrable functions $f:D\to \mathbb{C}$ (where $...
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How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$

Let $z\in \{z\in\mathbb{C}:|z|<1\}, \alpha>-1,\Gamma(s)$ is the gamma function. How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$ ? If $...
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Bergman space norm in terms of coefficients

I am interested in the Bergman space $A^2$ on the unit disc. According to the Wikipedia article on Bergman spaces, if we have $f(z)= \sum_{n=0}^\infty a_n z^n \in A^2$ then $$\|f\|^2_{A^2} := \frac{1}{...
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A continuous linear injection from $L^\infty$ into a separable subspace

Can there be a continuous linear injection of $L^\infty$ into one of its closed, separable subspaces? (Note: I am not requiring that injection to be surjective, nor to have closed range). ...
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Bergman Space: Analytic Sequence

In order to work with Bergman Spaces, I am trying to understand the next assertion Let $G$ be a complex domain. If $(f_n:G \to \mathbb{C})_n$ is a sequence of analytic functions that is uniformly ...
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Bergman space. What is area measure?

I have read that the Bergman space $A^p(\Omega)$ consist of all the analytic functions $f$ in $\Omega$, such that $$ \left( \int_{\Omega} |f(z)|^p dA \right)^{1/p} < \infty $$ where $dA$ is the ...
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Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
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Showing a subset of the Bergman space is closed

The following is problem 1.10 in chapter 1 of Conway's A Course in Functional Analysis. Let $G$ be an open subset of $\mathbb C$ and show that if $a\in G$, then $\{f\in L^2_a(G): f(a)=0\}$ is ...
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Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \...