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 $\backslash overline\{\backslash Omega\}$. 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

:$f\backslash mapsto\; Pf(z)$

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

:$Pf(z)\; =\; \backslash int\_\{\backslash partial\backslash Omega\}\; f(\backslash zeta)\backslash overline\{k\_z(\backslash zeta)\}\backslash ,d\backslash sigma(\backslash zeta).$

The Szegő kernel is defined by

:$S(z,\backslash zeta)\; =\; \backslash overline\{k\_z(\backslash zeta)\},\backslash quad\; z\backslash in\backslash Omega,\backslash zeta\backslash in\backslash partial\backslash Omega.$

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

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