Best L1 Approximation and Locally Computed L1 Spline Fits of the Heaviside Function and Multiscale Univariate Datasets

Best L1 approximations of the Heaviside function in Chebyshev and weak-Chebyshev spaces has a Gibbs phenomenon. It has been shown in the nineties for the trigonometric polynomial [1] and polygonal line cases [2]. By mean of recent results of characterization of best L1 approximation in Chebyshev and weak-Chebyshev spaces [3] that we recall, this Gibbs phenomenon can also be evidenced in the polynomial and polynomial spline cases. It can be reduced in this latter case by using L1 spline fits [4] which are best L1 approximations in an appropriate spline space obtained by the reunion of L1 interpolation splines [5]. These splines are known to preserve the shape of the Heaviside function [6]. We prove here the existence of L1 spline fits. Their major disadvantage is that obtaining them can be time consuming. Thus we propose a sliding window method on seven knots which is as efficient as the global method but within a linear complexity on the number of spline knots. This algorithm can also be fairly applied to the problem of approximation of datasets with abrupt changes of magnitude.