中文摘要：

目的 在实际问题中，某些插值问题结点处的函数值往往是未知的，而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先，利用积分值的线性组合来逼近结点处的函数值；然后，利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值；最后，给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数，一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。

英文摘要：

Objective The integral values of some successive equidistant subintervals are known in practical areas, whereas the usual function values at the knots are not given in interpolation problems. We propose a multilevel integro cubic spline quasi-interpolation for function approximation from given integral values over successive subintervals and multilevel spline quasi-interpolation.Methods We used the linear combination of the given integral values to approximate function values at knots. The multilevel cubic spline quasi-interpolation operator was defined with the classical cubic spline quasi-interpolation and its corresponding error function. Finally, we obtained its polynomial reproducing property and error estimate.Results The proposed method, together with the existing integro cubic spline quasi-interpolation, was tested by two infinitely differentiable functions. Numerical experiments showed that the proposed method possessed better approximation behaviors and numerical convergence orders compared with the integro cubic spline quasi-interpolation.Conclusion Multilevel integro cubic spline quasi-interpolation can successfully approximate the original function and its first and second-order derivative functions over the global interval. This process has good approximation behavior and numerical convergence compared with the existing integro spline quasi-interpolation. Moreover, the proposed method of function reconstruction from the integral values of successive subintervals is universally applicable.