最小支集样条小波插值函数当N=10的误差分析

Error Analysis About the Minimum Supported Spline WaveletInterpolation Function When N=10

讨论了当N=10时4阶(3次)B样条N4(x)对应的最小支集样条小波插值函数SΨ4(x)的影响,得到了当端点t0与t10处分别存在误差ε′0与ε′10点一阶导数和各分点存在误差时对SΨ4(t)=SΨ4(t+3.5)=SΨ4(x),详见文中式(15)～(17))的系数的改变量最大不超时,WΨ4(t)(其中WΨ4(t)的函数值、一阶导数值及二阶导数值过4.5h(|ε′0|+|ε′10|),在各分点ti/2,(i=0,1,…,20)处WΨ4的改变量最大分别不超过3.15h(|ε′0|+|ε′10|)、0.36(|ε′0|+|ε′10|)、144.23(|ε′0|+|ε′10|)/h;而当(t)的系数的改变量最大不超过各分点ti/2,(i=0,1,…,20)存在误差εi/2,(i=0,1,…,20)时WΨ4{|εi/2|}),在各分点ti/2,(0,1,…,20)处WΨ4(t)的函数值、一阶导数值及二阶导189.8ε(其中ε=max0≤i≤20数值的改变量最大分别不超过132.86ε、394.78ε/h、6077.40ε/h2.

The minimum supported spline wavelet interpolation function S_(Ψ_4)(x)-corresponding to the B-spline of the fourth order (three degree) N_4(x) is studied when nodes or the first derivate at the endpoints have errors as N=10. We have got the maximum change for the coefficient of W_(Ψ_4)(t)(W_(Ψ_4)(t)=S_(Ψ_4)(t+3.5)=S_(Ψ_4)(x)), which is no more than 4.5h(｜ε′_0｜+｜ε′_(10)｜) and the maximum change for the value of function and the first derivate and the second derivate are respectively no more than 3.15h((｜ε′_0｜)+(｜ε′_(10)｜)), 0.36(｜ε′_0｜+｜ε′_(10)｜), 144.23(｜ε′_0｜+(｜ε′_(10)｜))/h when the first derivates at t_0 and t_(10) have the errors ε′_0 and ε′_(10); and the maximum change for the coefficient of W_(Ψ_4)(t) is no more than 189.8ε and the maximum change for the value of function and the first derivate and, the second derivate are respectively no more than 132.86ε,394.78ε/h, 6 077.40ε/h^2 when the nodes t_(i/2) (i=0, 1, …,20) have the error ε_(i/2), (i=0,1,…,20)(where ε=(max0≤i≤20{｜ε_(i/2)｜})).