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Bergman space
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(Definition)
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Let $G \subset {\mathbb{C}}^n$ be a domain and let $dV$ denote the Euclidean volume measure on $G$
Definition 1 Let \begin{equation*} A^2(G) := \Big\{ f \text{ holomorpic in } G ~\Big|~ \sqrt{ \int_G \lvert f(z) \rvert^2 dV(z) } < \infty \Big\} . \end{equation*}$A^2(G)$ is called the Bergman space on $G$ The norm on this space is defined as \begin{equation*} \lVert f \rVert := \sqrt{ \int_G \lvert f(z) \rvert^2 dV(z) } . \end{equation*}Further we define an inner product on $A^2(G)$ as \begin{equation*} \langle f , g \rangle := \int_G f(z) \overline{g(z)} dV(z) . \end{equation*}
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 $L_a^2(G)$
- 1
- D'Angelo, John P. Several complex variables and the geometry of real hypersurfaces, CRC Press, 1993.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Bergman space" is owned by jirka.
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Cross-references: Hilbert space, compact subsets, uniform convergence, normal convergence, implies, complete, inner product, norm, euclidean volume measure, domain
There are 2 references to this entry.
This is version 7 of Bergman space, born on 2005-02-22, modified 2008-11-04.
Object id is 6801, canonical name is BergmanSpace.
Accessed 3656 times total.
Classification:
| AMS MSC: | 32A36 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Bergman spaces) |
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Pending Errata and Addenda
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