$L_1$ spline fits via sliding window process : continuous and discrete cases

TL;DR

This paper introduces an efficient sliding window method for $L_1$ spline fitting that reduces Gibbs phenomenon in approximating functions like the Heaviside and multiscale datasets, with proven existence of such fits.

Contribution

It proves the existence of $L_1$ spline fits and proposes a linear-complexity sliding window algorithm for their computation.

Findings

Reduces Gibbs phenomenon in $L_1$ spline approximations

Proposes a linear complexity sliding window method

Achieves efficiency comparable to global methods

Abstract

Best $L_1$ approximation of the Heaviside function and best $\ell_1$ approximation of multiscale univariate datasets by cubic splines have a Gibbs phenomenon. Numerical experiments show that it can be reduced by using $L_1$ spline fits which are best $L_1$ approximations in an appropriate spline space obtained by the union of $L_1$ interpolation splines. We prove here the existence of $L_1$ spline fits which has never been done to the best of our knowledge. Their major disadvantage is that obtaining them can be time consuming. Thus we propose a sliding window method on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude but within a linear complexity on the number of spline nodes.