在现实中, 某些结点处的函数值往往是未知的, 而在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的信息来解决函数重构是一个重要的问题. 文章首先从理论上证明了连续区间上积分值的偶次样条插值的存在唯一性. 其次, 我们给出了连续区间上积分值的偶次样条插值的光滑性质并且指出四次样条插值是最光滑的. 最后, 文章给出了偶次样条插值函数去逼近结点处的函数值和偶次高阶导数值时具有超收敛性的猜想. 这个猜想在随后的八次样条插值例子中得到证实.

In some practical areas, the integral values of some subintervals are known, whereas the usual function values at the knots are not given. It is an important problem to reconstruct the original function from integral values of successive intervals. In this paper, we firstly prove the existence and uniqueness of the even degree spline which interpolates given integral values of successive intervals. Secondly, we analyze the smoothness of such even degree spline and prove that the quartic spline is the smoothest interpolating function. Finally, we give the conjecture on super convergence of such even degree spline when it approximates the function values and even order derivatives at the knots. This conjecture is verified by the subsequent eight degree spline interpolation.