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 Ω ¯ {\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
f ↦ P f ( z ) {\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 kz, which is to say
P f ( z ) = ∫ ∂ Ω f ( ζ ) k z ( ζ ) ¯ d σ ( ζ ) . {\displaystyle Pf(z)=\int _{\partial \Omega }f(\zeta ){\overline {k_{z}(\zeta )}}\,d\sigma (\zeta ).}
The Szegő kernel is defined by
S ( z , ζ ) = k z ( ζ ) ¯ , z ∈ Ω , ζ ∈ ∂ Ω . {\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 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 , ζ ) = ∑ i = 1 ∞ ϕ i ( z ) ϕ i ( ζ ) ¯ . {\displaystyle 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
