Abstract
The best L 1 approximation of the Heaviside function and the best ℓ 1 approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L 1 spline fits which are the 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 for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm 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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Acknowledgments
The authors thank deeply Shu-Cherng Fang and Ziteng Wang from the Industrial and Systems Engineering Department of North Carolina State University and John E. Lavery, retired from the Army Research Office, for their comments and suggestions that improved the contents of this paper.
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Gajny, L., Gibaru, O. & Nyiri, E. L 1 spline fits via sliding window process: continuous and discrete cases. Numer Algor 78, 449–464 (2018). https://doi.org/10.1007/s11075-017-0383-0
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DOI: https://doi.org/10.1007/s11075-017-0383-0