Piecewise linear approximation of smooth functions of two variables

TL;DR

This paper demonstrates that smooth functions of two variables can be approximated by piecewise linear functions with controlled differential graph areas, providing a new method for approximation in geometric analysis.

Contribution

It introduces a novel approximation technique for smooth functions using piecewise linear functions with bounded differential graph areas in two dimensions.

Findings

Approximation of smooth functions by PL functions with area control

Construction of a unique polyhedron representing the differential graph

Universal constant bounds the area of differential graphs in approximation

Abstract

Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ representing the graph of the differential of $p$. Restricting to dimension 2, we show that any smooth function $f(x,y)$ may be approximated by a sequence $p_1,p_2,\dots$ of PL functions such that the areas of the $\D(p_i)$ are locally dominated by the area of the graph of $df$ times a universal constant.