Bergman space

In complex analysis, a branch of mathematics, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, $A^{p}(D)$ is the space of holomorphic functions in D such that the p-norm

$\|f\|_{p}=\left(\int _{D}|f(x+iy)|^{p}\,dx\,dy\right)^{1/p}<\infty .$

Thus $A^{p}(D)$ is the subspace of homolorphic functions that are in the space L^p(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:

$\sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{p}(D)}.$

(1)

Thus convergence of a sequence of holomorphic functions in L^p(D) implies also compact convergence, and so the limit function is also holomorphic.

If p = 2, then $A^{p}(D)$ is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.

References

• Richter, Stefan (2001) [1994], "Bergman spaces", Encyclopedia of Mathematics, EMS Press.