Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x...Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x） =（sin（n+1）θ）/（sinθ）（x=cosθ θ∈[0,π]）, the second kind Chebyshevpolynomical. All the symbols below have the same meaning as Ref. [1]if notspecifically defined. We shall consider a kind of new interpolating problem in thisnote. For any non-negative integer q and f∈C[-1, 1], it is well known that thepolynomial Q<sub>nq</sub>（f）∈П<sub>N</sub> （N=2（q+1） （n+1） -1） satisfying the following conditions isuniquely determined:Q<sub>nq</sub>(f, x<sub>kn</sub>) =f(x<sub>kn</sub>), 1≤k≤n; Q<sub>nq</sub>（f,±1）=f（±1）,Q<sub>nq</sub><sup>j</sup>(f,x<sub>kn</sub>)=c<sub>jkn</sub>, 1≤k≤n,1≤j≤2q+1,Q<sub>nq</sub><sup>j</sup>（f,1）=d<sub>jn</sub>, Q<sub>nq</sub><sup>j</sup>（f,-1）=g<sub>jn</sub>, 1≤j≤q,where c<sub>jkn</sub>,d<sub>jn</sub>, g<sub>jn</sub>are any given real numbers. Q<sub>nq</sub>（f）is called the higher orderquasi Hermite-Fejer interpolation of f.We展开更多
Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one...Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.展开更多
文摘Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x） =（sin（n+1）θ）/（sinθ）（x=cosθ θ∈[0,π]）, the second kind Chebyshevpolynomical. All the symbols below have the same meaning as Ref. [1]if notspecifically defined. We shall consider a kind of new interpolating problem in thisnote. For any non-negative integer q and f∈C[-1, 1], it is well known that thepolynomial Q<sub>nq</sub>（f）∈П<sub>N</sub> （N=2（q+1） （n+1） -1） satisfying the following conditions isuniquely determined:Q<sub>nq</sub>(f, x<sub>kn</sub>) =f(x<sub>kn</sub>), 1≤k≤n; Q<sub>nq</sub>（f,±1）=f（±1）,Q<sub>nq</sub><sup>j</sup>(f,x<sub>kn</sub>)=c<sub>jkn</sub>, 1≤k≤n,1≤j≤2q+1,Q<sub>nq</sub><sup>j</sup>（f,1）=d<sub>jn</sub>, Q<sub>nq</sub><sup>j</sup>（f,-1）=g<sub>jn</sub>, 1≤j≤q,where c<sub>jkn</sub>,d<sub>jn</sub>, g<sub>jn</sub>are any given real numbers. Q<sub>nq</sub>（f）is called the higher orderquasi Hermite-Fejer interpolation of f.We
基金Supported by the National Natural Science Foundation of China (Grant No. 10601065)
文摘Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.
