两类Chebyshev零点的Newman型有理算子逼近|x|的渐近性质

The asymptotic property of approximation to |x| by Newman-type rational operators at the two kinds of Chebyshev nodes

设X={xk∶k=1,2,…,n}是区间(0,1]上n个互不相同点的集合,令pn(x)=∏nk=1(xk+x),rn(X;x)=xpn(x)-pn(-x)pn(x)+pn(-x),本文给出了当X=U={xk=cosk2n+1π∶k=1,2…,n},X=T={xk=sin2k-14nπ∶k=1,2,…,n}时,max|x|≤1‖x|-rn(U;x)|及max|x|≤1‖x|-rn(T;x)|的渐近表达式.

LetX= {xk ：k= 1,2,…,n} bea set of n distinct points in（0,1]. Denote byp.（x） =∏nk=1（xk＋x）,rn（X;x）=xpn（x）-pn（-x）/pn（x）＋pn（＋x）,The present note gives the asymptotic representations of max｜x｜≤1‖x｜-rn（U;x）｜ and maxx≤1‖x｜-rn（T;x）｜ when X=U={xk=cosk/2n＋1π∶k=1,2…,n} and X=T={xk=sin 2k-1/4nπ∶k=1,2,…,n}.