  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.
