高精度积分值四次B样条拟插值

High Accuracy Integro Quartic B-Spline Quasi-Interpolation

摘要: 目的：在函数重构问题中，节点处的函数值是未知的，而连续区间上的积分值是已知的。如何利用连续区间上积分值信息来解决函数重构是一个重要的问题。本文给出一种高精度积分值四次B样条拟插值算子解决连续区间上积分值的函数重构问题。方法：首先利用连续区间上的积分值的线性组合构造新的泛函系数，即结点处的函数值和一阶导数值的逼近值，进而基于带导数信息的高精度B样条拟插值算子，得到高精度积分值四次B样条拟插值。最后给出该算子对函数及其高阶导数的逼近误差。结果：在数值实例中，从逼近误差、数值收敛阶等方面验证了该方法的有效性。结论：本文构造的算子不依赖于目标函数导数信息，不需额外边界信息，不需求解方程组并且次数为四次，计算更为简便。与已有成果相比较，对函数及其高阶导数的逼近精度更高。

Abstract: Objective: In the function reconstruction problem, the function values at the knots are unknown, and the integral values on the successive subintervals are known. How to use the integral values information on the successive subintervals to solve the function reconstruction is an important problem. In this paper, the high accuracy integro quartic B-spline quasi-interpolation operator given in this paper effectively solves the problem of function reconstruction of integral values on successive subintervals. Method: Firstly, a new functional coefficient is constructed by using the linear combination of the integral values on the successive subintervals, that is, the function values at the knots and the approximation value of the first-order derivative. Then, based on the high accuracy B-spline quasi-interpolation operator with derivative information, the high accuracy integro quartic B-spline quasi-interpolation is obtained. Finally, the approximation error of the operator to the function and its higher order derivatives is given. Result: In numerical examples, the effectiveness of the method is verified from the aspects of approximation error and numerical convergence order. Conclusion: The operator constructed in this paper does not depend on the derivative information of the objective function, does not need additional boundary information, does not need to solve the equations, and the number of times is four, and the calculation is simpler. Compared with the existing results, the approximation accuracy of the function and its higher order derivatives is higher.