$L_1$ spline fits via sliding window process : continuous and discrete cases
For researchers in approximation theory and signal processing, this work provides a theoretical foundation and efficient algorithm for $L_1$ spline fitting, though it is incremental as it extends existing $L_1$ interpolation concepts.
The paper proves the existence of $L_1$ spline fits for reducing Gibbs phenomena in cubic spline approximations, and proposes a sliding window method that achieves linear complexity while matching the global method's performance on functions and datasets with abrupt changes.
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.