Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

This paper establishes the following pointwise result for simultaneous Lagrange interpolating approximation: Let f∈C _?1,1] ^q and r=q+2/2], then $$\left| {f^{(k)} (x) - P_n^{(k)} (f,x)} \right| = O(1)\Delta _n^{q - k} (x)\omega (f^{(q)} ,\Delta _n (x))(||L_n || + ||L_n $$ where P_n(f,x) is the Lagrange interpolating polynomial of degree n+2r...1 of f(x) on the nodes X_n∪Y_n (see the definition of the next), $\Delta _n (x) = \frac{{N1 - x2}}{n} + \frac{1}{{n^2 }}$ .