精确华宁不等式与最佳 Hermite 插值结点组

利用Hermite插值误差的余项估计式，在最大框架下确定了Sobolev空间在最大和平均范数下逼近问题的最优Hermite插值结点组，并对在Hermite插值结点组上Hermite数据为零的函数给出了计算华宁不等式最佳常数的方法。先利用构造辅助函数的方法给出了Hermite插值误差的一种估计式，在此基础上把华宁不等式最佳常数的计算转化为一个显式积分表达式，并用两个例子来说明结果。同时在最大框架下给出了Sobolev空间利用Hermite插值逼近误差的准确值，并找出了当Hermite插值结点个数固定时的最优Hermite插值结点组。对一些特殊情形，给出了最优插值结点组的显式表达式；对于一般情形，把最优插值结点组的计算归结为求一些具体函数的最小值点。利用Mathematical计算了华宁不等式最优系数的一些具体值。

Sharp Wirtinger Inequalities and Optimal Hermite Interpolation Nodes

In the worst case setting, by using the remainder estimate of the Hermite interpolation, the optimal Hermite interpolation nodes for the approximation problem of the Sobolev spaces in maximal and mean norms, respectively, are determined. The method for calculating the best constants in the Wirtinger's inequality is given for functions whose Hermite data is vanished at Hermite interpolation nodes. First, a remainder estimate of the Hermite interpolation approximation error is given by using the method of constructing auxiliary function. After that, the calculation of the best constants in the Wirtinger's inequality is turned into an explicit integral expression, and two examples are used to illustrate the results. At the same time, in the worst case setting, the approximation error values of the Sobolev spaces by using Hermite interpolation are given, and the optimal Hermite interpolation nodes are found when the number of Hermite interpolation nodes is fixed. For some special cases, the explicit expression for the optimal interpolation nodes is given. For the general case, the calculation of the optimal interpolation nodes is reduced to finding the minimum point of some specific functions. Using Mathematical, the values of some optimal coefficients in the Wirtinger's inequality are obtained.