Bergman space
In complex analysis, functional analysis and operator theory, 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, for , the Bergman space  is the space of all holomorphic functions  in D for which the p-norm is finite:

The quantity  is called the norm of the function ; it is a true norm if . Thus  is the subspace of holomorphic functions that are in the space Lp(D). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:

Thus convergence of a sequence of holomorphic functions in  implies also compact convergence, and so the limit function is also holomorphic. If , then  is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel. Special cases and generalisations

If the domain  is bounded, then the norm is often given by:

where  is a normalised Lebesgue measure of the complex plane, i.e. . Alternatively  is used, regardless of the area of . The Bergman space is usually defined on the open unit disk  of the complex plane, in which case . In the Hilbert space case, given:, we have:

that is,  is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular the polynomials are dense in . Similarly, if , the right (or the upper) complex half-plane, then:

where , that is,  is isometrically isomorphic to the weighted Lp1/t (0,∞) space (via the Laplace transform). The weighted Bergman space  is defined in an analogous way, i.e.,

provided that  is chosen in such way, that  is a Banach space (or a Hilbert space, if ). In case where , by a weighted Bergman space  we mean the space of all analytic functions  such that:

and similarly on the right half-plane (i.e., ) we have:

and this space is isometrically isomorphic, via the Laplace transform, to the space , where:

(here  denotes the Gamma function). Further generalisations are sometimes considered, for example  denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure  on the closed right complex half-plane , that is:

Reproducing kernels

The reproducing kernel  of  at point  is given by:

and similarly, for  we have:

In general, if  maps a domain  conformally onto a domain , then:

In weighted case we have:

and:

References

Further reading 

. See also

Bergman kernel
Banach space
Hilbert space
Reproducing kernel Hilbert space
Hardy space
Dirichlet space

Complex analysis
Functional analysis
Operator theory