A Note on Gradually Varied Functions and Harmonic Functions

TL;DR

This paper explores the relationship between harmonic functions and gradually varied functions, showing that many harmonic functions are nearly gradually varied, which has implications for approximation and discretization in continuous functions.

Contribution

It demonstrates that many harmonic functions are near gradually varied, connecting harmonic analysis with the concept of gradual variation in discretized domains.

Findings

Many harmonic functions are near gradually varied.

Gradually varied functions are generally not harmonic.

The study relates harmonic functions to gradual variation properties.

Abstract

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we get some interesting properties related to gradual variation? In this note, we try to prove that many harmonic functions are gradually varied or near gradually varied; this means that the value of the center point differs from that of its neighbor at most by 2. It is obvious that most of the gradually varied functions are not harmonic.This note discusses some of the basic harmonic functions in relation to gradually varied functions.