We consider the problem on shape-preserving interpolation by classical cubic splines. Namely, we consider conditions guaranteeing that, for a positive function (or a function whose kth derivative is positive), the cubic spline (respectively, its kth derivative) is positive. We present a survey of known results, completely describe the cases in which boundary conditions are formulated in terms of the first derivative, and obtain similar results for the second derivative. We discuss in detail mathematical methods for obtaining sufficient conditions for shape-preserving interpolation. We also develop such methods, which allows us to obtain general conditions for a spline and its derivative to be positive. We prove that, for a strictly positive function (or a function whose derivative is positive), it is possible to find an interpolant of the same sign as the initial function (respectively, its derivative) by thickening the mesh.

中文翻译：

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三次样条插值的保形条件

我们考虑经典三次样条在形状保持插值上的问题。也就是说，我们考虑条件，对于正函数（或第k个导数为正的函数），保证三次样条曲线（分别为其k三阶导数）为正。我们提供了一个已知结果的调查，完全描述了根据一阶导数制定边界条件的情况，并获得了与二阶导数相似的结果。我们将详细讨论为保持形状插值获得足够条件的数学方法。我们还开发了这样的方法，使我们能够获得样条及其导数为正的一般条件。我们证明，对于严格正函数（或其导数为正的函数），可以通过加厚网格找到与初始函数（分别为其导数）符号相同的插值。