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 [math]\displaystyle{ \overline{\Omega} }[/math]. 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

[math]\displaystyle{ f\mapsto Pf(z) }[/math]

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

[math]\displaystyle{ Pf(z) = \int_{\partial\Omega} f(\zeta)\overline{k_z(\zeta)}\,d\sigma(\zeta). }[/math]

The Szegő kernel is defined by

[math]\displaystyle{ S(z,\zeta) = \overline{k_z(\zeta)},\quad z\in\Omega,\zeta\in\partial\Omega. }[/math]

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

[math]\displaystyle{ S(z,\zeta) = \sum_{i=1}^\infty \phi_i(z)\overline{\phi_i(\zeta)}. }[/math]

References

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