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

Best $L_1$ approximation of the Heaviside function and best ${\mathrm{\ell }}_1$ 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.