The problem of interpolation by quartic splines according to Marsden’s scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in L₁-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden’s scheme for any function from the class C¹ on an arbitrary sequence of grids without any constraints. In Marsden’s scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin’s scheme (the data grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in L_∞-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class C³.

中文翻译：

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四次插值样条曲线的收敛

考虑了根据Marsden方案的四次样条插值问题。示出了根据内插样条的二阶导数在L ₁归一化的二次B样条中的扩展系数的计算，产生了针对所选参数的线性方程组。该系统的矩阵是五对，并且具有一列对角优势，这使得它可以有效地计算所需的参数，建立根据马斯登的用于从类中的任何函数方案的样条内插过程的收敛Ç ¹在没有任何约束的任意网格序列上。在Marsden的方案中，假定给出了一个结网格，并且严格在中间选择了插值节点。根据Subbotin的方案，已建立的结果通过四次样条传递到插值的情况（数据网格和结网格互换）。这里方程在第三衍生物的膨胀系数的系统大号_∞ -normalized B样条具有对角优势，以及用于从类中的任何内插功能的内插处理收敛Ç ³。