Extension of functions with bounded finite differences

TL;DR

This paper proves that functions with bounded finite differences on a lattice in a finite-dimensional torus can be smoothly extended to the entire torus, establishing a relationship between bounds on derivatives and finite differences.

Contribution

It introduces a method to extend functions with bounded finite differences on a lattice to smooth functions on the torus, linking finite difference bounds to derivative bounds.

Findings

Functions with bounded finite differences can be smoothly extended to the entire torus.

The bounds on the extension's derivatives are related to the original finite difference bounds.

Abstract

We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original function's finite differences.