Bergman space

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

Definition.

Let

A^{2}(G):=\Big{\{}f\text{ holomorpic in }G~{}\Big{|}~{}\sqrt{\int_{G}\lvert f(% z)\rvert^{2}dV(z)}<\infty\Big{\}}.

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

\lVert f\rVert:=\sqrt{\int_{G}\lvert f(z)\rvert^{2}dV(z)}.

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

\langle f,g\rangle:=\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 $L_{a}^{2}(G)$ .

References

1 D’Angelo, John P. , CRC Press, 1993.
2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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