在实际问题中, 某些插值点处的函数值往往是未知的, 而仅仅已知一些连续等距区间上的积分值. 如何利用连续区间上积分值信息来解决函数重构是一个有意义的问题. 首先, 文章利用连续等距区间上的积分值信息直接构造了一类二次样条拟插值, 它称之为积分值型二次样条拟插值. 然后, 给出了积分值型二次样条拟插值的多项式再生性和逼近节点处函数值的超收敛性. 最后, 给出了一类改进的积分值型二次样条拟插值及其性质. 实验结果表明, 与已有的积分值型三次样条拟插值相比, 文章提出的拟插值更简单和有效, 并且可以推广到积分值型高次样条拟插值.

In some practical fields, the usual function values at the interpolated points are not known, whereas the integral values of some successive intervals are given. How to use the integral values of successive intervals to solve function reconstruction is a significant problem. This paper firstly proposes a class of quadratic spline quasi-interpolants from the integral values of successive intervals. It is a direct construction and it is called integro quadratic spline quasi-interpolants. Secondly, we analyze its polynomial reproducing property and give its super convergence property when it approximates the function values at the knots, and so on. Lastly, a type of modified integro quadratic spline quasi-interpolants and its corresponding properties are presented. Experiments show that the proposed quasi-interpolants are simpler and more effective than the existing integro cubic spline quasi-interpolants. Moreover, it can be generalized to integro spline quasi-interpolants with higher degree.