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A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials

Part of: Approximations and expansions Numerical methods in Fourier analysis Numerical approximation and computational geometry

Published online by Cambridge University Press:  09 December 2019

A. Pepin ,
S. S. Beauchemin ,
S. Léger  and
N. Beaudoin
A. Pepin
Affiliation:
Département de mathématiques et de statistique, Pavillon Rémi-Rossignol, Université de Moncton, 18 avenue Antonine-Maillet, Moncton, Canada, E1A 3E9 Email: eap0070@umoncton.casophie.leger@umoncton.ca
S. S. Beauchemin
Affiliation:
Department of Computer Science, University of Western Ontario, Middlesex College 28C, London, Canada, N6A 5B7 Email: sbeauche@uwo.ca
S. Léger
Affiliation:
Département de mathématiques et de statistique, Pavillon Rémi-Rossignol, Université de Moncton, 18 avenue Antonine-Maillet, Moncton, Canada, E1A 3E9 Email: eap0070@umoncton.casophie.leger@umoncton.ca
N. Beaudoin
Affiliation:
Département de physique et d’astronomie, Pavillon Rémi-Rossignol, Université de Moncton, 18 avenue Antonine-Maillet, Moncton, Canada, E1A 3E9 Email: normand.beaudoin@umoncton.ca
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Abstract

Effective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.

In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.

Keywords

splinecontinuityFFTDFTdiscrete Fourier transformnumerical derivative

MSC classification

Primary: 41A15: Spline approximation
Secondary: 65D05: Interpolation 65D07: Splines 65T50: Discrete and fast Fourier transforms
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 655 - 669
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

We would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the New Brunswick Innovation Foundation (NBIF), the Université de Moncton and Assumption Life.

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