# Bergman kernel

Let $G\subset{\mathbb{C}}^{n}$ be a domain (http://planetmath.org/Domain2). And let $A^{2}(G)$ be the Bergman space. For a fixed $z\in G$, the functional $f\mapsto f(z)$ is a bounded linear functional. By the Riesz representation theorem (as $A^{2}(G)$ is a Hilbert space) there exists an element of $A^{2}(G)$ that represents it, and let’s call that element $k_{z}\in A^{2}(G)$. That is we have that $f(z)=\langle f,k_{z}\rangle$. So we can define the .

###### Definition.

The function

 $K(z,w):=\overline{k_{z}(w)}$

is called the Bergman kernel.

By definition of the inner product in $A^{2}(G)$ we then have that for $f\in A^{2}(G)$

 $f(z)=\int_{G}f(w)K(z,w)dV(w),$

where $dV$ is the volume measure.

As the $A^{2}(G)$ space is a subspace of $L^{2}(G,dV)$ which is a separable Hilbert space then $A^{2}(G)$ also has a countable orthonormal basis, say $\{\varphi_{j}\}_{j=1}^{\infty}$.

###### Theorem.

We can compute the Bergman kernel as

 $K(z,w)=\sum_{j=1}^{\infty}\varphi_{j}(z)\overline{\varphi_{j}(w)},$

where the sum converges uniformly on compact subsets of $G\times G$.

Note that integration against the Bergman kernel is just the orthogonal projection from $L^{2}(G,dV)$ to $A^{2}(G)$. So not only is this kernel reproducing for holomorphic functions, but it will produce a holomorphic function when we just feed in any $L^{2}(G,dV)$ function.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Bergman kernel BergmanKernel 2013-03-22 15:04:45 2013-03-22 15:04:45 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32A25 BergmanSpace BergmanMetric