Bergman kernel


Let Gn be a domain (http://planetmath.org/Domain2). And let A2(G) be the Bergman space. For a fixed zG, the functional ff(z) is a boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath linear functionalMathworldPlanetmath. By the Riesz representation theorem (as A2(G) is a Hilbert spaceMathworldPlanetmath) there exists an element of A2(G) that represents it, and let’s call that element kzA2(G). That is we have that f(z)=f,kz. So we can define the Bergman kernelMathworldPlanetmath.

Definition.

The function

K(z,w):=kz(w)¯

is called the Bergman kernel.

By definition of the inner productMathworldPlanetmath in A2(G) we then have that for fA2(G)

f(z)=Gf(w)K(z,w)𝑑V(w),

where dV is the volume measure.

As the A2(G) space is a subspaceMathworldPlanetmathPlanetmath of L2(G,dV) which is a separable Hilbert space then A2(G) also has a countable orthonormal basisMathworldPlanetmath, say {φj}j=1.

Theorem.

We can compute the Bergman kernel as

K(z,w)=j=1φj(z)φj(w)¯,

where the sum converges uniformly on compact subsets of G×G.

Note that integration against the Bergman kernel is just the orthogonal projection from L2(G,dV) to A2(G). So not only is this kernel reproducing for holomorphic functionsMathworldPlanetmath, but it will produce a holomorphic function when we just feed in any L2(G,dV) 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