Green's functions and discrete demi-analytic kernels

Abstract

Both Hilbert spaces of discrete analytic functions and Hilbert spaces of discrete harmonic functions are studied in this paper. Discrete analogs of the partial/partial z and partial/partial z-bar operators and a new line integral are defined and used to establish the relationship between the discrete Green's function and the Bergman kernel for the Hilbert space of discrete analytic functions on a discrete region. Similar relations between the discrete Green's and Neumann's functions and the reproducing kernel for the Hilbert space of discrete harmonic functions on a rectangle are also discussed. These results are used to establish certain types of convergence of the discrete harmonic kernel to the harmonic kernel. The space of discrete harmonic functions is then complexified and the convergence properties of the discrete harmonic kernel are applied to the real part of the discrete Bergman kernel for the space of discrete analytic functions on a rectangle.