Representation and Superposition of Discrete Functions and Equations with Parameterized Operations

TL;DR

This paper presents constructive solutions for discrete equations using superpositions of discrete functions with limited values, extending classical results for continuous functions to discrete settings with parameterized operations.

Contribution

It introduces a method to solve discrete equations via superpositions of discrete functions, including new operators for parameterized operations, extending classical continuous function results.

Findings

Solutions for discrete equations using superpositions of discrete functions.

Introduction of four special operators for parameterized operations.

Extension of results to discrete operators and operator equations.

Abstract

Existence results for Hilbert's problem 13th mean that any equation constructed by continue functions can be given solution represented as a superposition of continue functions of one variable or of continue functions of two variables. Constructive results for discrete functions are given in this paper that any equation constructed by functions called discrete 3 function for which field of definition is a set containing only -1,0,1 can be given solution represented as a superposition of discrete 3 functions of one variable or of two variables. Formula solution for equation with parameterized operations can be given after introducing four special operators being correspondence among known operations and new operations. These results can be extended to discrete operators and operator equations constructed by them.