Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S29 - S64|
|Published online||26 February 2021|
On a class of splines free of Gibbs phenomenon
Department of Applied Mathematics and Statistics, Universidad Politécnica de Cartagena, Cartagena, Spain
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
* Corresponding author: email@example.com
Accepted: 28 March 2020
When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.
Mathematics Subject Classification: 65D05 / 65D17 / 65M06 / 65N06
Key words: Splines / adaption to discontinuities / interpolation / computer aided design (modeling of curves)
© EDP Sciences, SMAI 2021
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