关于二元函数的三角插值逼近

Approximation by bivariate trigonometric interpolation polynomials

目的为克服Lagrange插值多项式不能对任意连续函数都一致收敛的问题,构造了一类二元乘积型三角插值多项式算子使得该算子在全平面上能够一致收敛到每个以2π为周期的二元连续函数。方法通过对Lagrange插值三角多项式的平移与组合,在已有成果的基础上做了推广,构造了一类形式较为广泛的二元乘积型三角插值多项式Tmn(f;x,y)=sum from κ=0 to 2m sum from l=0 to 2n f(xκ,yl)mακ(x)mβl(x),进而讨论了该算子的逼近性质。结果/结论证明了该算子在全平面上一致收敛到任意以2π为周期的二元连续函数,并且对C2sπ,r,2π(s≤α,r≤β)函数类的逼近均达到最佳收敛阶,即,当f(x,y)∈Cs2,πr,2π,s≤α,r≤β,成立|Tmn(f;x,y)-f(x,y)|=O{Em*n(f)+1/m^sω(~sf/x^s;1/m,0)+r/n^1ω(~rf/y^r;0,1/n)+1/m^s 1/n^rω(^(s+r)f/x^sy^r;1/m,1/n)}。

Aim To avoid the Lagrange operator can not converge uniformly to any continuous functions, a kind of bivariate product operator of trigonometric interpolation polynomials is constructed. And this kind of operator can converge uniformly to every bivariate continuous function of 27 periodic on total plane. Methods As a generalization of previous results, a kind of more general bivariate product operator of trigonometric interpolation polynomials Tmn （f;x, y） =∑k=0^2m∑l=0^2nf（xk,yl）mα^k（x）mβ^l（x）is constructed. Furthermore, the approximation properties of the operator are researched. Results/Conclusion It＇s proved that the operator can converge uniformly to any bivariate continuous function of 2π periodic on total plane, and the convergence order is the best, i. e.if f（x,y）∈C2π,2π^s,r,s≤α,r≤β,｜Tmn（f;x,y）-f（x,y）｜=O{Emn^＊（f）＋1/m^sω（ ^sf/ x^s;1/m,0）＋1/n^rω（ ^rf/ y^r;0,1/n）＋1/m^s1/n^rω（ ^s＋rf/ x^s y^r;1/m,1/n）}.