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 set of all holomorphic functions in Ω that are continuous on \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

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 kz, 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 φi 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)}.

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