A reproducing kernel function and convergence properties for discrete analytic functions

Abstract

In 1944, J. Ferrand introduced the concept of a discrete analytic function. Many properties of discrete analytic functions were developed by R. J. Duffin in 1956. In the first part of this dissertation a reproducing kernel is defined and is shown to have analogous properties of the Bergman kernel for analytic functions. The next part of the paper discusses various types of convergence of discrete analytic functions to a given analytic function defined on a simply connected domain, Finally, it is shown that it is not reasonable to expect the discrete kernels to converge to the Bergman kernel defined on a simply connected domain.