On discrete analytic functions: Products, Rational Functions, and some Associated Reproducing Kernel Hilbert Spaces

TL;DR

This paper introduces a new family of discrete analytic functions, defines two products within this family, and explores their properties, including rational functions and connections to Schur analysis in a specialized function space.

Contribution

It develops the concept of expandable discrete analytic functions, introduces two novel products, and establishes a framework for rational functions and Schur analysis in this context.

Findings

Defined a family of discrete analytic functions including polynomials

Introduced two products enabling the definition of rational functions

Explored Schur analysis within a new space of entire functions

Abstract

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.