On simultaneous approximation by lagrange interpolating polynomials

Abstract

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

$$|f^{\left( k \right)} \left( x \right) - P^{\left( k \right)} \left( {f,x} \right)| = O\left( 1 \right)\Delta _n^{q - k} \left( x \right)\omega \left( {f^{\left( q \right)} ,\Delta \left( x \right)} \right)\left( {||L_n || + ||L_n * * ||} \right),0 \leqslant k \leqslant q,$$

(1)

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.