摘要
This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.
This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.
