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 C with "C" boundary, and let "A"(Ω) denote the space of all holomorphic functions in Ω that are continuous on formula_1. Define the Hardy space "H"(∂Ω) to be the closure in "L"(∂Ω) of the restrictions of elements of "A"(Ω) to the boundary. The Poisson integral implies that each element "ƒ  of H"(∂Ω) extends to a holomorphic function "Pƒ" in Ω. Furthermore, for each "z" ∈ Ω, the map defines a continuous linear functional on "H"(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel "k", 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 "φ" is an orthonormal basis of "H"(∂Ω) consisting entirely of the restrictions of functions in "A"(Ω), then a Riesz–Fischer theorem argument shows that