L₁ spline fits via sliding window process : continuous and discrete cases

2015/10/26 by Gajny, Laurent, Gibaru, Olivier, Nyiri, Eric
#65D07 #65D10 #65K10 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1510.07557

Abstract

Best L₁ approximation of the Heaviside function and best ℓ₁ approximation of multiscale univariate datasets by cubic splines have a Gibbs phenomenon. Numerical experiments show that it can be reduced by using L₁ spline fits which are best L₁ approximations in an appropriate spline space obtained by the union of L₁ interpolation splines. We prove here the existence of L₁ 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.

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