Bergman space

Let $G\subset {ℂ}^n$ be a domain and let $dV$ denote the Euclidean volume measure on $G$.

Definition.

Let

$A^2(G)$ is called the Bergman space on $G$. The norm on this space is defined as

$$\parallel f\parallel :=\sqrt{{\int }_G{|f(z)|}^2𝑑V(z)}.$$

Further we define an inner product on $A^2(G)$ as

$$⟨f,g⟩:={\int }_{G}f(z)\overline{g(z)}𝑑V(z).$$

The inner product as defined above really is an inner product and further it can be shown that $A^2(G)$ is complete since convergence in the above norm implies normal convergence (uniform convergence on compact subsets). The space $A^2(G)$ is therefore a Hilbert space. Sometimes this space is also denoted by $L_a^2(G)$.

Title	Bergman space
Canonical name	BergmanSpace
Date of creation	2013-03-22 15:04:43
Last modified on	2013-03-22 15:04:43
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	10
Author	jirka (4157)
Entry type	Definition
Classification	msc 32A36
Related topic	BergmanKernel