In this paper a special class of one-dimensional L -splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L -splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions permits a simple and direct numerical implementation. Applications are provided to financial data of the index S&P500, for the fast calculation of statistically interesting quantities, as cross validation (scores), generalized cross validation (scores) for finding the best smoothing parameter $$\alpha $$ α , and the residual sum of squares.

中文翻译：

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一类 4 阶 L 样条：用于插值和平滑的快速算法

本文研究了一类特殊的 4 阶一维 L 样条，它自然出现在用多元多样条进行插值和平滑的计算中。提供了使用此类 L 样条进行插值和平滑的快速算法，以及 Reinsch 算法对此设置的推广。所有数学表达式的明确描述允许简单和直接的数值实现。为指数 S&P500 的财务数据提供了应用程序，用于快速计算统计上有趣的数量，作为交叉验证（分数）、用于寻找最佳平滑参数的广义交叉验证（分数）$$\alpha $$ α 和残差平方和。