Best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces

TL;DR

This paper investigates the best $L_1$ approximation of Heaviside functions within Chebyshev and weak-Chebyshev spaces, extending theoretical results and applying them to polynomial and spline approximations.

Contribution

It extends the Hobby-Rice theorem framework and proves the uniqueness of best $L_1$ approximation in specific Chebyshev spaces, including even-dimensional cases.

Findings

Extended Hobby-Rice theorem for Chebyshev spaces

Proved uniqueness of best $L_1$ approximation under certain conditions

Applied results to polynomial and spline approximation of Heaviside functions

Abstract

In this article, we study the problem of best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem into an appropriate framework and prove the unicity of best $L_1$ approximation of Heaviside-type functions in an even-dimensional Chebyshev space under the condition that the dimension of the subspace composed of the even functions is half the dimension of the whole space. We also apply the results to compute best $L_1$ approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.