We express the interpolating cubic splines of class C2 in their new, explicit forms. We construct the desired forms, the spline's Hermitian and B-spline representations for both equidistant and arbitrary nodes. During this process we demonstrate an innovative way to compute the inverse of a special class of tridiagonal matrices. Afterward, we propose the corresponding interpolating spline based linear regression models with easily interpretable coefficients suitable for smoothing data of complex structures.

中文翻译：

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插值三次样条和数据平滑的显式形式


我们以 C2 类的插值三次样条的新显式形式表示它们。我们构建所需的形式，即等距和任意节点的样条曲线和 B 样条曲线表示。在此过程中，我们展示了一种计算特殊类三对角矩阵的逆的创新方法。然后，我们提出了相应的基于样条的插值线性回归模型，这些模型具有易于解释的系数，适用于对复杂结构的数据进行平滑处理。