Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linear formulas. The order of approximation, in each case, is calculated and several numerical experiments are performed to contrast the theoretical results.

中文翻译：

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具有强梯度的函数的单调三次样条插值

由于样条插值在插值的平滑度和准确性方面具有良好的特性，因此已在多种应用中使用。但是，当数据中存在不连续或陡峭的梯度时，由于吉布斯现象会出现一些伪像。同样，在某些应用程序中必须保持数据单调性，并且插值器不会自动验证该属性。在本文中，我们研究了基于Hermite三次插值器获得单调三次样条的充分条件，并提出了使用非线性公式构造它们的不同方法。计算每种情况下的近似阶数，并进行一些数值实验以对比理论结果。