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 L<sup>p</sup>(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.
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 âÂÂ<sup>p</sup>(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 (or the upper) complex half-plane, then:
where , that is, is isometrically isomorphic to the weighted L<sup>p</sup><sub>1/t</sub> (0,âÂÂ) 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:
It is possible to generalise to the (weighted) 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 (reproducing kernel) Hilbert space when .
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 (respectively other Bergman spaces that are RKHS), it is true that