Szegő kernel

From Wikipedia, the free encyclopedia

In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő.

Let Ω be a bounded domain in C^{n with C² boundary, and let A(Ω) denote the set of all holomorphic functions in Ω that are continuous on . Define the Hardy space H²(∂Ω) to be the closure in L²(∂Ω) of the restrictions of elements of A(Ω) to the boundary. The Poisson integral implies that each element ƒ of H²(∂Ω) extends to a holomorphic function Pƒ in Ω. Furthermore, for each z ∈ Ω, the map}

$f\mapsto Pf(z)$

defines a continuous linear functional on H²(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel k_z, which is to say

$Pf(z)=\int _{{\partial \Omega }}f(\zeta )\overline {k_{z}(\zeta )}\,d\sigma (\zeta ).$

The Szegő kernel is defined by

$S(z,\zeta )=\overline {k_{z}(\zeta )},\quad z\in \Omega ,\zeta \in \partial \Omega .$

Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in z. In fact, if φ_i is an orthonormal basis of H²(∂Ω) consisting entirely of the restrictions of functions in A(Ω), then a Riesz–Fischer theorem argument shows that

$S(z,\zeta )=\sum _{{i=1}}^{\infty }\phi _{i}(z)\overline {\phi _{i}(\zeta )}.$

References

Krantz, Steven G. (2002), Function Theory of Several Complex Variables, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2724-6

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.