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.

中文翻译：

---

一种新的高阶样条插值方法：分段多项式的连续性证明

在当今的应用中，有效且准确的高度样条插值仍然是一项具有挑战性的任务。更高阶样条插值并不常用，因为它需要知道给定网格上函数节点处的高阶导数。在本文中，我们的目标是证明分段多项式及其在连接点处的导数的连续性，这是通过 Beaudoin (1998, 2003) 和 Beauchemin (2003) 最初开发的方法获得的。这种涉及离散傅立叶变换 (DFT/FFT) 的新方法可以为区间 [0, T ] 上的等距数据实现更高阶的样条插值。为此，我们分析了在求解方程组时可能出现的奇异点，这些方程组能够构造任何阶的样条。我们还注意到奇次样条和偶次样条之间的重要区别。这些结果证明了 Beaudoin 和 Beauchemin 的方法导致了任何程度的样条插值，并且这种新方法最终可以用于提高传统问题中样条插值的精度。