目的 在现实中,某些插值问题结点处的函数值往往是未知的,而仅仅已知一些区间上的积分值。为此提出一种给定已知函数在连续等距区间上的积分值构造二次样条插值函数的方法。方法 首先,利用二次B样条基函数的线性组合去满足给定的积分值和两个端点插值条件,该插值问题等价于求解n+2个方程带宽为3的线性方程组。然后,运用算子理论给出二次样条插值函数的误差估计,继而得到二次样条函数逼近结点处的函数值时具有超收敛性。最后,通过等距区间上积分值的线性组合逼近两个端点的函数值方法实现了不带任何边界条件的积分型二次样条插值问题。结果 选取低频率函数,对积分型二次样条插值方法和改进方法分别进行数值测试,发现这两种方法逼近效果都是良好的。同样,选取高频率函数对积分型二次样条插值方法进行数值实验,得到数值收敛阶与理论值相一致。结论 实验结果表明,本文算法相比已有的方法更简单有效,对改进前后的二次样条插值函数在逼近结点处的函数值时的超收敛性得到了验证。该方法对连续等距区间上积分值的函数重构具有普适性。

Objective In several practical interpolation problems, the integral values of several subintervals are known, whereas the usual function values at the knots are not given. In this article, we propose an integro quadratic spline interpolation method from the integral values of successive subintervals. Method First, we use the linear combination of quadratic B-spline functions to satisfy the given integral values and two additional boundary conditions. This problem is equivalent to solving a system of n+2 linear equations with a three-band coefficient matrix. Second, we use operator theory to conduct an error analysis and obtain the super convergence order in approximating the function values at the knots. Lastly, we deal with the integro interpolation problem even without any boundary condition based on the linear combination of integral values to approximate the boundary function values. Result The proposed and modified methods are tested by functions with low frequency in Example 1; the approximation behavior are satisfactory. Meanwhile, the proposed method is tested by functions with high frequency in Example 2; the numerical convergence order is consistent with the theoretical value. Conclusion The proposed method is easier to implement than other existing methods. The super convergence in approximating function values at the knots is also verified. The proposed method of function reconstruction from the integral values of successive subintervals is universally applicable.