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Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulas

机译：积分差分多项式和三角形样条和正交公式

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摘要

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.
机译：这项工作是众所述之一，该众所述是进一步调查第五阶近似的局部内插多项式样条。这里，提出了新的多项式和三角基本样条。这些样条键的主要特征如下：近似是针对每个网格间隔（或基本矩形）单独构造的近似，构造为基本样条的产品和和节点中的功能值和/或值的函数的近似它的衍生品和/或此功能的积分价值在子宫内。通过使用由一组功能提供的求解方程来确定基本样条。众所周知，当通过间隔内的功能的积分等于通过间隔内的函数的近似的积分时，近似具有一些物理并行。这里构造的花键满足第五级近似的性质。这里，当函数的值在每个插值中已知的函数是已知的时，构建第五阶近似的一维多项式和三相的基本样条。为了施工样条曲线，我们使用第一个衍生物和正交的离散模式以适当的近似顺序。我们将这些样条的属性与样条曲线进行比较，当函数的第一导数的值在每个插值中都是已知的，并且给出了每个网格间隔的积分值。通过使用张量产品样条构造，一维壳体可以扩展到多个尺寸。表示数值例子。

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