The discrete Szëgo kernel

Abstract

Of concern on this paper are complex-valued functions defined on the integer lattice (i.e. the set ) which are discrete analytic according to the definition given by Ferrand. In particular, we will study a Hilbert space consisting of the boundary values of discrete analytic functions defined on a finite simply connected union of unit squares of the integer lattice (a simple region), which is a discrete version of the Szëgo space. We will prove that this space admits a reproducing kernel, the discrete Szëgo kernel and will develop a general method to construct it. To sum up, the main merit of this paper is to present by means of an orthogonal projection operator a way to select among boundary values, those that can be extended to an analytic continuation.