On simultaneous approximation by lagrange interpolating polynomials

This paper considers to replace $\Delta _n \left( x \right) = \frac{{\sqrt {1 - x^2 } }}{n} + \frac{1}{{n^2 }}$ in the following result for simultaneous Lagrange inter polating approximation with $\sqrt {1 - x^2 } /n$ : Let $ \in C_{\left { - 1,1} \right]}^a $ and $r = \left {\frac{{q + 2}}{2}} \right]$ then 1 $$|f^{\left( k \right)} \left( x \right) - P^{\left( k \right)} \left( {f,x} \right)| = O\left( 1 \right)\Delta _n^{q - k} \left( x \right)\omega \left( {f^{\left( q \right)} ,\Delta \left( x \right)} \right)\left( {||L_n || + ||L_n * * ||} \right),0 \leqslant k \leqslant q,$$ where P_n(f,x) is the Lagrange inter polating polynomial of degree n+2r?1 of f on the nodes X_n ∪ Y_n (see the definition of the text), and thus give a problem raised in XiZh] a complete answer.