Bergman kernel

In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cⁿ.

In detail, let L²(D) be the Hilbert space of square integrable functions on D, and let L^2,h(D) denote the subspace consisting of holomorphic functions in D: that is,

$L^{2,h}(D)=L^{2}(D)\cap H(D)$

where H(D) is the space of holomorphic functions in D. Then L^2,h(D) is a Hilbert space: it is a closed linear subspace of L²(D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D

$\sup _{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{2}(D)}$ (1)

for every compact subset K of D. Thus convergence of a sequence of holomorphic functions in L²(D) implies also compact convergence, and so the limit function is also holomorphic.

Another consequence of (1) is that, for each z ∈ D, the evaluation

$\operatorname {ev} _{z}:f\mapsto f(z)$

is a continuous linear functional on L^2,h(D). By the Riesz representation theorem, this functional can be represented as the inner product with an element of L^2,h(D), which is to say that

$\operatorname {ev} _{z}f=\int _{D}f(\zeta ){\overline {\eta _{z}(\zeta )}}\,d\mu (\zeta ).$

The Bergman kernel K is defined by

$K(z,\zeta )={\overline {\eta _{z}(\zeta )}}.$

The kernel K(z,ζ) holomorphic in z and antiholomorphic in ζ, and satisfies

$f(z)=\int _{D}K(z,\zeta )f(\zeta )\,d\mu (\zeta ).$

See also

• Bergman metric
• Bergman space
• Szegő kernel

References

• Krantz, Steven G. (2002), Function Theory of Several Complex Variables, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2724-6.
• Chirka, E.M. (2001) [1994], "Bergman kernel function", Encyclopedia of Mathematics, EMS Press.