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 &Omega; be a bounded domain in Cn with C2 boundary, and let A(&Omega;) denote the space of all holomorphic functions in &Omega; that are continuous on $$\overline{\Omega}$$. Define the Hardy space H2(∂&Omega;) to be the closure in L2(∂&Omega;) of the restrictions of elements of A(&Omega;) to the boundary. The Poisson integral implies that each element &fnof; of H2(∂&Omega;) extends to a holomorphic function P&fnof; in &Omega;. Furthermore, for each z &isin; &Omega;, the map
 * $$f\mapsto Pf(z)$$

defines a continuous linear functional on H2(∂&Omega;). By the Riesz representation theorem, this linear functional is represented by a kernel kz, which is to say
 * $$Pf(z) = \int_{\partial\Omega} f(\zeta)\overline{k_z(\zeta)}\,d\sigma(\zeta).$$

The Szegő kernel is defined by
 * $$S(z,\zeta) = \overline{k_z(\zeta)},\quad z\in\Omega,\zeta\in\partial\Omega.$$

Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in z. In fact, if &phi;i is an orthonormal basis of H2(∂&Omega;) consisting entirely of the restrictions of functions in A(&Omega;), then a Riesz–Fischer theorem argument shows that
 * $$S(z,\zeta) = \sum_{i=1}^\infty \phi_i(z)\overline{\phi_i(\zeta)}.$$