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Questions tagged [bergman-spaces]

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 $\mathbb{C}$

A Bergman space, named after Stefan Bergman, is a function space of analytic functions in a domain $G$ of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. In a formal way, we define the Bergman space $A^p(G)$ as $$ A^p(G)=\left\{ f:G\to \mathbb{C} : f \text{ is analytic on $G$ and }\int_{G} |f(z)|^p dA < \infty \right\} $$ Where $dA$ is the area measure. It can be shown that $A^p(G)$ is a closed subspace of $L^p(G)$ and hence a Banach space itself.

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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)…
fbg
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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 between functional analysis, operator theory and…
Squirtle
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Show that a function is in Bergman space

Consider the Bergman space $A^2(\mathbb{D})$, where $\mathbb{D}$ is the open unit disk in $\mathbb{C}$. Show that if $\sum_{n=0}^\infty |a_n|^2$ converges, then $\sum_{n=0}^\infty (n+1)^{1/2} a_n z^n$ is holomorphic in $\mathbb{D}$. I want to show…
stedmoaoa
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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}{\pi} \int_\mathbb{D} |f(z)|^2 \, dz =…
Math Maniac
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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 area measure. I am very confuse with what area measure…
Leo Sera
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Convergence of Fourier series in Bergman norm

Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $0< p<+\infty$, define: $$a^p_\nu(D):=h(D)\cap…
Bob
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How to show that the set of polynomials is dense?

Consider a norm on the space of analytic functions $\|f\|=\int_{\mathbb C} \frac{|f|}{e^{|z|}} \, dz$. Is the set of polynomials dense in the space of analytic functions with bounded norm? My guess is yes, but it is really hard to show. If we have…
sam
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Problem in computing a sum.

For a multi-index $\alpha = \left (\alpha_1, \cdots, \alpha_n \right )$ we write $\left \lvert \alpha \right \rvert = \alpha_1 + \cdots + \alpha_n$ and $\alpha! = \alpha_1! \cdots \alpha_n!.$ For $z = (z_1, \cdots, z_n) \in \mathbb C^n$ and…
Anacardium
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Bergman Kernel for $L_p$ space (p $\neq$ 2)

For a given space $\mathcal{X}$, consider the Banach space $L_p(\mathcal{X},\mu)$ for the measure space $(\mathcal{X},\mu)$. I'm trying to understand the Bergman kernel for $L_p$ when $p \neq 2$. For $p = 2$, for the set of holomorphic functions…
user224369
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Orthonormal Basis of Bergman Space

This is a problem (#39 or #40 depending on the edition) at the end of Chapter 1 in Krantz's book Function Theory of Several Complex Variables. Let $\Omega\subset\mathbb{C}^n$ be a smooth and bounded domain. Let $z_0\in\Omega$. Goal: To construct an…
t-rex
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The Carleson measure on Bergman space.

Let $\mu$ be positive Borel measure on unit disk D. We say that $\mu$ is a Carleson measure on the Bergman space if there is a constant $C> 0$ such that $f\in A^2(D,dA)$, $$\left\|f\right\|_\mu^2 := \int_D|f(z)|^2d\mu\leq C…
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Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman reproducing kernel. Let $\varphi_{\lambda}$ be…
Anacardium
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If the Toeplitz operator $T_u = 0$ then $u = 0$ almost everywhere.

Let $u \in L^{\infty} (\mathbb D).$ Assume that $T_u = 0,$ where $T_u$ is the Toeplitz operator on $A^2 (\mathbb D)$ corresponding to the symbol $u.$ Then $T_u (f) = 0$ for every $f \in A^2 (\mathbb D)$ and hence $P (uf) = 0,$ where $P : L^2…
Akiro Kurosawa
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calculating Bergman kernel of punctured unit disk

Any idea how can we calculate the Bergman kernel of the punctured unit disk? I know how to find the Bergman kernel of the unit disk, for example, we have complete orthonormal basis for the Bergman space $A^2(\mathbb{D})$. Is it true that the…
Adam_math
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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 closed in $L^2_a(G)$. Conway defines the space…
Potato
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