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 in D for which the p-norm is finite:

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

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, we have

that is, is isometrically isomorphic to the weighted ℓp(1/(n + 1)) space. In particular, not only are the polynomials dense in .

Similarly, if , the right (or the upper) complex half-plane, then:

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 a Banach space (or a Hilbert space, if we mean the space of all analytic functions such that:

) we have:

, where:

Here denotes the Gamma function.

denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure, that is:

to the (weighted) Bergman space of vector-valued functions,.

Reproducing kernels

of is given by:

we have:

, then:

In weighted case we have:

and:

In any reproducing kernel Bergman space, functions obey a certain property. It is called the reproducing property.

References

Further reading

• .

See also

• Bergman kernel

• Banach space

• Hilbert space

• Reproducing kernel Hilbert space

• Hardy space

• Dirichlet space