Exponentials form a basis of discrete holomorphic functions

TL;DR

This paper demonstrates that discrete exponential functions constitute a basis for discrete holomorphic functions, and on convex domains, discrete polynomials also form a basis, advancing the understanding of discrete complex analysis.

Contribution

It establishes the foundational role of discrete exponentials and polynomials as bases in discrete holomorphic function spaces.

Findings

Discrete exponentials form a basis for discrete holomorphic functions.

Discrete polynomials form a basis on convex domains.

Provides a new framework for discrete complex analysis.

Abstract

We show that discrete exponentials form a basis of discrete holomorphic functions. On a convex, the discrete polynomials form a basis as well.