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 and also 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 of the space Lp(D). The Bergman spaces are Banach spaces for, which is a consequence of the following estimate that is valid on compact subsets K of D:Convergence of a sequence of holomorphic functions in thus implies 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. If, given an element, we have
that is, is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular, not only are the polynomials dense in, but every function can be uniformly approximated by radial dilations of functions holomorphic on a disk, where and the radial dilation of a function is defined by for.
Similarly, if, the right complex half-plane, then:
where, that is, is isometrically isomorphic to the weighted Lp1/t (0,∞) space.
The weighted Bergman space is defined in an analogous way, i.e.,
provided that is chosen in such way, that is a Banach space. 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 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 with respect to a translation-invariant positive regular Borel measure on the closed right complex half-plane, that is:
It is possible to generalise to the Bergman space of vector-valued functions, defined byand the norm on this space is given asThe measure is the same as the previous measure on the weighted Bergman space over the unit disk, is a Hilbert space. In this case, the space is a Banach space for and a Hilbert space when.
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:
In any reproducing kernel Bergman space, functions obey a certain property. It is called the reproducing property. This is expressed as a formula as follows: For any function, it is true that