Approximation of curves with piecewise constant or piecewise linear functions

TL;DR

This paper analyzes how well piecewise constant and linear functions approximate continuous curves by computing the Hausdorff distance within Sobolev norm-based sets, linking discretization, smoothing, and combinatorial mathematics.

Contribution

It introduces methods to measure approximation quality of curves using Hausdorff distance within Sobolev spaces and explores the connection between B-splines, discretization, and Eulerian numbers.

Findings

Computed Hausdorff distances for curve approximations

Established discretization and smoothing procedures preserving Sobolev norms

Linked B-splines and Eulerian numbers in the context of curve approximation

Abstract

In this paper we compute the Hausdorff distance between sets of continuous curves and sets of piecewise constant or linear discretizations. These sets are Sobolev balls given by the continuous or discrete $L^p$-norm of the derivatives. We detail the suitable discretization or smoothing procedure which are preservative in the sense of these norms. Finally we exhibit the link between Eulerian numbers and the uniformly space knots B-spline used for smoothing.