Error bounds for Lagrange interpolation (Q1344171)

Consider an interpolation of functions \(f\in W_ \infty^ m [a,b]\) by Lagrange polynomials \(\ell_{m-1}\), \(\Delta(f)\) of degree \(m-1\) at the mesh \(\Delta\) of the interpolating nodes \(\{t_ j\}^ m_ 1\). Error bounds due to this approximation is evaluated as \[ L_{m,k} (\Delta)= \sup_{x\in [a,b]} L_{m,k} (\Delta,x)= {\textstyle {1\over m!}} \omega_ \Delta^{(k)} (\cdot), \] for all \(m\) and \(k\) \((0\leq k\leq m- 1)\), and for any mesh of the interpolating nodes.