Bergman kernel
Let G⊂ℂn be a domain (http://planetmath.org/Domain2). And let A2(G) be the Bergman space. For a fixed z∈G, the functional f↦f(z) is a bounded
linear functional
. By the Riesz representation theorem (as A2(G) is a Hilbert space
) there exists an element of A2(G) that represents it, and
let’s call that element kz∈A2(G). That is we have that
f(z)=⟨f,kz⟩. So we can define the Bergman kernel
.
Definition.
As the space is a subspace of which is a separable Hilbert space then also has a countable orthonormal basis
, say .
Theorem.
Note that integration against the Bergman kernel is just the orthogonal
projection from to . So not only is this kernel reproducing for holomorphic functions, but it will produce a holomorphic function when we just feed in any function.
References
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title | Bergman kernel |
---|---|
Canonical name | BergmanKernel |
Date of creation | 2013-03-22 15:04:45 |
Last modified on | 2013-03-22 15:04:45 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 7 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32A25 |
Related topic | BergmanSpace |
Related topic | BergmanMetric |