中文摘要：

数值微分是用离散的函数值近似地求出函数在某点的导数值,此问题在阿达马（Hadamard）意义下是一个不适定问题,即在测量过程中的微小误差可能造成数值结果的巨大误差。用磨光化方法构造了数值微分问题的正则解,给出误差估计。理论分析和实验证明,此方法可以用来寻找函数的间断点,并可应用于Abel积分方程的误差估计。

英文摘要：

Numerical differentiation problem is to approximately evaluate the function＇ s derivative value at a certain point by discrete function value. This problem in the Hadamard＇ s meaning is an ill-posed problem, namely, small errors during the measurement may result in large errors in the numerical results. Mollification method is used to construct the regular solution of numerical differentiation, and offer the error estimate. It is indicated from both theoretical and experimental aspects that the mollification method can be used to find the discontinuous point of function, as well as to get the error estimate of Abel integral equation.