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 denote the set of all holomorphic functions in Ω that are continuous on. Define the Hardy space H2 to be the closure in L2 of the restrictions of elements of A to the boundary. The Poisson integral implies that each element ƒ of H2 extends to a holomorphic function Pƒ in Ω. Furthermore, for each z ∈ Ω, the map
defines a continuous linear functional on H2. By the Riesz representation theorem, this linear functional is represented by a kernel kz, which is to say
The Szegő kernel is defined by
Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in z. In fact, if φi is an orthonormal basis of H2 consisting entirely of the restrictions of functions in A, then a Riesz–Fischer theorem argument shows that
