新二元三角插值多项式的构造

设Ω=-πxπ,-πyπ],C(Ω)表示关于x,y均以2π为周期的连续函数空间.若f(x,y)∈C(Ω),取结点组为(xk,yl)=(2k+2n 1)π,(2l 2+m 1)πk=0,1,2,…,2n,l=0,1,2,…,2m,则我们获得一个二元三角插值多项式Cn,m(f;x,y)=M1N∑k=2n0∑l=2m0f(xk,yl).1+2∑nα=1cosα(x-xk)+2∑mβ=1cosβ(y-yl)+4∑nα=1∑mβ=1cosα(x-xk)cosβ(y-yl)其中M=2m+1,N=2n+1.为改进其收敛性,本文构造一个新的因子ρα,β,使得带有该因子ρα,β的二元三角插值多项式Ln,m(f;x,y)可以在全平面上一致地收敛到每个连续的f(x,y),且具有最佳逼近阶.

Structuer of a New Double Trigonometric Interpolation Polynomials

Let Ω=-π≤x≤π,-π≤y≤π],and C(Ω) be the continuous function space of the periodic function with period 2π.if the function f(x,y)∈C(Ω),at nodes as follows:(xk,yl)=(2k+1)π2n,(2l+1)π2mk=0,1,2,…,2n,l=0,1,2,…,2m;Therefore,we obtain a new double trigonometric interpolation polynomial Cn,m(f;x,y)=1MN∑2nk=0∑2ml=0f(xk,yl)1+2∑nα=1cos α(x-xk) +2∑mβ=1cos β(y-yl)+4∑nα=1∑mβ=1cos α(x-xk)cos β(y-yl)where M=2m+1,N=2n+1We construct a summation factor ρα,β in order to improve its convergence order,such that the integral operator Ln,m(f;x,y) with the factor ρα,β convergence uniformly on total plane for any f(x,y)∈C(Ω),and has the best approximation order.