Bergman space

Special cases and generalisations

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

:|f|_{A^p(D)} := \left(\int_D |f(z)|^p,dA\right)^{1/p} ; ; ; ; ; (f \in A^p(D)),

where A is a normalised Lebesgue measure of the complex plane, i.e. . Alternatively is used, regardless of the area of D. The Bergman space is usually defined on the open unit disk \mathbb{D} of the complex plane, in which case A^p(\mathbb{D}):=A^p. If p=2, given an element f(z)= \sum_{n=0}^\infty a_n z^n \in A^2, we have

:|f|^2_{A^2} := \frac{1}{\pi} \int_\mathbb{D} |f(z)|^2 , dz = \sum_{n=0}^\infty \frac{|a_n|^2}{n+1},

that is, A2 is isometrically isomorphic to the weighted ℓ**p(1/(n + 1)) space. In particular, not only are the polynomials dense in A2, but every function f \in A^2 can be uniformly approximated by radial dilations of functions g holomorphic on a disk D_R(0), where R 1 and the radial dilation of a function is defined by g_r(z) := g(rz) for 0 .

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

:|F|^2_{A^2(\mathbb{C}+)} := \frac{1}{\pi} \int{\mathbb{C}_+} |F(z)|^2 , dz = \int_0^\infty |f(t)|^2\frac{dt}{t},

where F(z)= \int_0^\infty f(t)e^{-tz} , dt, that is, A^2(\mathbb{C}_+) is isometrically isomorphic to the weighted L**p1/t (0,∞) space (via the Laplace transform).

The weighted Bergman space A**p(D) is defined in an analogous way, i.e.,

:|f|_{A^p_w (D)} := \left( \int_D |f(x+iy)|^2 , w(x+iy) , dx , dy \right)^{1/p},

provided that w : D → 0, ∞) is chosen in such way, that A^p_w(D) is a [Banach space (or a Hilbert space, if ). In case where D= \mathbb{D}, by a weighted Bergman space A^p_\alpha we mean the space of all analytic functions f such that:

: |f|{A^p\alpha} := \left( (\alpha+1)\int_\mathbb{D} |f(z)|^p , (1-|z|^2)^\alpha dA(z) \right)^{1/p}

and similarly on the right half-plane (i.e., A^p_\alpha(\mathbb{C}_+)) we have:

: |f|{A^p\alpha(\mathbb{C}+)} := \left( \frac{1}{\pi}\int{\mathbb{C}_+} |f(x+iy)|^p x^\alpha , dx , dy \right)^{1/p},

and this space is isometrically isomorphic, via the Laplace transform, to the space L^2(\mathbb{R}+, , d\mu\alpha), where:

:d\mu_\alpha := \frac{\Gamma(\alpha+1)}{2^\alpha t^{\alpha+1}} , dt.

Here Γ denotes the Gamma function.

Further generalisations are sometimes considered, for example A^2_\nu denotes a weighted Bergman space (often called a Zen space) with respect to a translation-invariant positive regular Borel measure \nu on the closed right complex half-plane \overline{\mathbb{C}_+}, that is:

:A^p_\nu := \left{ f : \mathbb{C}+ \longrightarrow \mathbb{C} \text{ analytic} ; : ; |f|{A^p_\nu} := \left( \sup_{\varepsilon0} \int_{\overline{\mathbb{C}+}} |f(z+\varepsilon)|^p , d\nu(z) \right)^{1/p} It is possible to generalise A^2 to the (weighted) Bergman space of vector-valued functions, defined byA^{2}\alpha(\mathbb{D}; \mathcal{H}) := \left{, f \colon , \mathbb{D} \to \mathcal{H} ; | ;f \text{ analytic and } |f|{2, \alpha} and the norm on this space is given as|f|{2, \alpha} = \left(\int_{\mathbb{D}} |f(z)|{\mathcal{H}}^2d\mu\alpha(z)\right)^{\frac{1}{2}}.The measure \mu_\alpha is the same as the previous measure on the weighted Bergman space over the unit disk, \mathcal{H} is a Hilbert space. In this case, the space is a Banach space for 0 and a (reproducing kernel) Hilbert space when p=2.

Reproducing kernels

The reproducing kernel k_z^{A^2} of A2 at point z \in \mathbb{D} is given by:

: k_z^{A^2}(\zeta)=\frac{1}{(1-\overline{z}\zeta)^2} ; ; ; ; ; (\zeta \in \mathbb{D}),

and similarly, for A^2(\mathbb{C}_+) we have:

: k_z^{A^2(\mathbb{C}+)}(\zeta)=\frac{1}{(\overline{z}+\zeta)^2} ; ; ; ; ; (\zeta \in \mathbb{C}+),

In general, if \varphi maps a domain \Omega conformally onto a domain D, then:

:k^{A^2(\Omega)}z (\zeta) = k^{\mathcal{A}^2(D)}{\varphi(z)}(\varphi(\zeta)) , \overline{\varphi'(z)}\varphi'(\zeta) ; ; ; ; ; (z, \zeta \in \Omega).

In weighted case we have:

:k_z^{A^2_\alpha} (\zeta) = \frac{\alpha+1}{(1-\overline{z}\zeta)^{\alpha+2}} ; ; ; ; ; (z, \zeta \in \mathbb{D}),

and:

:k_z^{A^2_\alpha(\mathbb{C}+)} (\zeta) = \frac{2^\alpha(\alpha+1)}{(\overline{z}+\zeta)^{\alpha+2}} ; ; ; ; ; (z, \zeta \in \mathbb{C}+). 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 f \in A^2 (respectively other Bergman spaces that are RKHS), it is true thatf(z) = \langle f, k_z^{A^2} \rangle_{2} = \int_{\mathbb{D}} \frac{f(\zeta)}{(1-z\overline{\zeta})^2}dA.

References

References

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