中文摘要：

光滑函数在光滑支持向量机的理论中起着重要作用．1996年Chen等人提出一个支持向量机的光滑函数———Sigmoid函数的积分函数，并解决了该光滑函数的误差问题．2005-2009年，袁玉波、熊金志和刘叶青等人相继提出支持向量机的无穷多个多项式光滑函数和多项式光滑的支持向量机模型，但都未解决这类多项式光滑函数的误差函数问题．为此，用 Newton‐Hermite 插值方法研究该问题．研究结果表明：1）用 New ton‐Hermite插值方法可计算这类光滑函数的误差函数，并给出了具体算法；2）这类误差函数有无穷多个，可用一个一般形式表示，并得到了这个一般形式；3）这类误差函数具有许多重要性质，并给出了严格证明．解决了支持向量机无穷多个多项式光滑函数的误差函数及其性质问题，建立了这类多项式光滑函数的误差理论，为研究支持向量机的光滑理论提供了基本的理论支持．

英文摘要：

Smoothing functions play an important role in the theory of smooth support vector machines .In 1996 ,Chen et al proposed a smoothing function of support vector machines—the integral function of Sigmoid function ,and solved the error problem of the smoothing function .From 2005 to 2009 ,Yuan ,Xiong and Liu proposed an infinite number of polynomial smoothing function and the corresponding reformulations for support vector machines .However ,they did not touch the error functions for this class of polynomial smoothing functions .To fill up this gap ,this paper studies the problem of the error functions with the Newton‐Hermite interpolation method .The results show that ：1） the error functions of this class of polynomial smoothing functions can be calculated using the Newton‐Hermite interpolation method ,and the detailed algorithm is given ;2） there are an infinite number of error functions for this class of polynomial smoothing functions and a general formulation is obtained to describe these error functions ;3） there are several important properties for this class of error functions and the strict proof is given for these properties .By solving the problem of the error functions and their properties , this paper establishes an error theory of this class of polynomial smoothing functions ,w hich is a basic theoretical support for smooth support vector machines .