EXPONENTIALS FORM A BASIS OF DISCRETE HOLOMORPHIC FUNCTIONS

EXPONENTIALS FORM A BASIS OF DISCRETE HOLOMORPHIC FUNCTIONS by Christian Mercat Abstract. — We show that discrete exponentials form a basis of discrete holomorphic functions on a critical map. On a combinatorially convex set, the discrete polynomials form a basis as well. Resume (Les exponentielles forment une base des fonctions holomorphes discretes) Nous montrons que les exponentielles forment une base des fonctions holomorphes discretes sur une carte critique. Sur un convexe, les polynomes discrets forment egalement une base. 1. Introduction The notion of discrete Riemann surfaces has been defined in [7]. A good basis for the associated space of holomorphic functions was still missing. This article discuss an interesting one in the simply connected critical case. We are interested in a cellular decomposition ? of the complex plane or a simply connected portion U of it, by rhombi (equilateral quadrilaterals, or lozenges). In other words, we have a map from the set of vertices ?0 to the complex plane Z : ?0 ? C such that for each oriented face (x, y, x?, y?) ? ?2, its image is a positively oriented rhombus (Z(x), Z(y), Z(x?), Z(y?)) of side length ? > 0. It defines a straightforward Cauchy-Riemann equation for a function f ? C0(?) of the vertices, and similarly for 2000 Mathematics Subject Classification.

• agree point-wise


• connected critical


• discrete exponentials


• converging discrete


• wise multiplication


• cauchy-riemann equation