Szegő kernel

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 Cⁿ with C² boundary, and let A(Ω) denote the space of all holomorphic functions in Ω that are continuous on ${\displaystyle {\overline {\Omega }}}$. 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

${\displaystyle f\mapsto Pf(z)}$

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

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

The Szegő kernel is defined by

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

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

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