逐次有理L_2逼近的收敛性

Convergence of successive rational L_2-approximation

设Rn(x)∈Rlm={P(x)/Q(x)},(n=1,2,…)是函数f(x)的第n次最佳L2逼近元,记Sn(x)=∑nk=1Rk(x),(n=1,2,…),在某些附加条件下证明了序列{Sn(x)}一致收敛于f(x),给出了序列{Sn(x)}一致收敛于f(x)的充要条件,并在另一较弱条件下证明了序列{Rn(x)}及其各阶导函数序列{R(k)n(x)},(k=1,2,…)一致收敛于零。

Let Rn(x)∈R^lm={P(x)/Q(x)}(n=1,2,…) be the nth best L2-approximation to f(x) and (Sn(x))=∑nk=1Rk(x)(n=1,2,…).It is proved that {Sn(x)} converges on f(x) uniformly under some (conditions). The necessary and sufficient condition that {Sn(x)} converges on f(x) is given. It is also proved that {Rn(x)} and its derivatives {R^((k))n(x)}(k=1,2,…) converge on zero uniformly under a weaker condition.