L 1 spline fits via sliding window process: continuous and discrete cases
- 48 Downloads
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.
KeywordsBest approximation L1 norm Shape preservation Polynomial spline Heaviside function Sliding window
Unable to display preview. Download preview PDF.
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.
- 5.Gajny, L., Gibaru, O., Nyiri, E., Fang, S.-C.: Best L 1 approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. Numer. Algorithms 75(3), 827–843 (2017)Google Scholar
- 6.Gajny, L., Nyiri, E., Gibaru, O.: Fast polynomial spline approximation for large scattered data sets via L 1 minimization. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, volume 8085 of Lecture Notes in Computer Science, pp 813–820. Springer, Berlin (2013)Google Scholar
- 11.Nürnberger, G.: Approximation by Spline Functions. Springer (1989)Google Scholar
- 14.Pinkus, A.: On L 1-approximation. Cambridge tracts in mathematics. Cambridge University Press, Cambridge (1989). On t.p. 1 is superscriptxGoogle Scholar