In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
,
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...
, a Bergman space, named after
Stefan Bergman, is a
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s in a
domain ''D'' of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
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
L''p''(''D''). The Bergman spaces are
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s, which is a consequence of the estimate, valid on
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is ...
, 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
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose d ...
of the complex plane, in which case
. In the Hilbert space case, given:
, we have:
:
that is, is isometrically isomorphic to the weighted
''ℓ''''p''(1/(''n'' + 1)) space.
In particular the
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s are
dense in . Similarly, if , the right (or the upper) complex half-plane, then:
:
where
, that is, is isometrically isomorphic to the weighted
''L''''p''1/''t'' (0,∞) space (via the
Laplace transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
).
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 mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
(or a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, 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
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
).
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
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
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
*
*
*{{springer, title=Bergman spaces, id=B/b120130, first=Stefan, last=Richter.
See also
*
Bergman kernel
*
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
*
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
*
Reproducing kernel Hilbert space
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is ...
*
Hardy space
*
Dirichlet space
Complex analysis
Functional analysis
Operator theory