一类有理插值算子的逼近

On the Approximation by Rational Interpolation Operators

文本讨论了以Legendre多项式及其导数的零点为结点的一类有理逼近,得到了它的精确点态估计.

For f∈C[-1,1], let ω(f,δ) be the modulus of continuity of f and Hω= {f:ω(f,δ)≤ω((δ)}, where ω(δ) is a given modulus of continuity. Let Pn(x) be the n-th Legendre polynomial with normalization Pn(1) = l, and let -1 = y2n+1<y2n<…<y2<y1=1 be the zeros of (1-x2)Pn(x)P'n(x). In this paper, we consider the rational interpolatior operators based on {yk}where lk(x) is the basic polynomial of Lagrange interpolation based on the nodes z. The main result is the following.Theorem 1. For s>0, the following relationholds uniformly on [-1,1], where and y, satisfy the conditions