Let be a domain (http://planetmath.org/Domain2). And let be the Bergman space. For a fixed , the functional is a bounded linear functional. By the Riesz representation theorem (as is a Hilbert space) there exists an element of that represents it, and let’s call that element . That is we have that . So we can define the Bergman kernel.
is called the Bergman kernel.
By definition of the inner product in we then have that for
where is the volume measure.
We can compute the Bergman kernel as
where the sum converges uniformly on compact subsets of .
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.
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Date of creation||2013-03-22 15:04:45|
|Last modified on||2013-03-22 15:04:45|
|Last modified by||jirka (4157)|