【摘要】 取Gn(f,x)为以Legendre多项式零点为节点的Grunwald插值多项式。本文证明了对连续函数f(x),Gn(f,x)在开区间(-1,1)上处处收敛到f(x),并得到了Gn(f,x)逼近f(x)的阶。最后得到的主要结果表明,对于全实轴上任何增长型的连续函数总可被全实轴上扩展了的Grunwald插值多项式几乎处处一致逼近。更多还原

【Abstract】 Let Gn (f, x) be.polynomial of Grunwald interpolation, whichis based on the zeros of the Legendre polynomial.In this paper,it is provedthat Gn(f,x) converges everywhere to f(x)∈C[-1,1] in the open interval[-1,1],and obtained that the order of approximation for Gn(f,x)to tend to f(x) is0[1/4 (1/2)n+?(f,1/3(1/2)n)]. Finally,the main result obtained is that a continuousfunction of any type of growth on the interval (-∞,∞) can always be almostuniformly approximated by a kind of extended grunwald interpolation polyno-mial defined on (-∞, ∞).更多还原