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

• Published in: Canadian mathematical bulletin Vol. 63; no. 3; pp. 655 - 669
• Main Authors: Pepin, A., Beauchemin, S. S., Léger, S., Beaudoin, N.
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical Society 01.09.2020; Cambridge University Press
• Subjects: Continuity; Derivatives; Fourier transforms; Interpolation; Mathematics; Polynomials; Singularity (mathematics); Spline functions; 65T50; numerical derivative; DFT; FFT; 65D05; continuity; spline; 41A15; 65D07; discrete Fourier transform
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439519000742

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.