Convergence of Lagrange interpolation polynomials for piecewise smooth functions (Q1603252)

The authors study Lagrange interpolation on the nodes \[ x_k^{(n)}=\cos{{k-1\over n-1}\pi},\;1\leq k\leq n, \] the zeros of \(\omega_n(x)=\sqrt{1-x^2}\sin{((n-1)\arccos{x})}\) for piecewise smooth functions on \([-1,1]\) and their main result is an estimate for the error in terms of the modulus of continuity on the subintervals and the location of the nodes. Consider a function \(f\in\text{C}^r\text{C}^{r+1}\) i.e. 1. \(f\in\text{C}^r\) on \([-1,1]\), 2. there exists a partition \(-1=a_{s+1}<a_s<\cdots <a_1<a_0=1\) with: 3. \(f\in\text{C}^{r+k}\) on \([a_{j+1},a_j],\;0\leq j\leq s\) Put \[ A_n(x)=\min \Biggl\{1,{1\over n\min_{1\leq j\leq s} |x-a_j|}\Biggr\}, \] and let \(\omega(f^{(r+k)},\delta)_j\) be the modulus of continuity of \(f^{(r+k)}\) on the subinterval \([a_{j+1},a_j],\;0\leq j\leq s\). Then: \[ |f(x)-L_n(x)|=O\left({|\omega_n(x)|\over n^{r+1}}\right) \left[A_n(x)+\max_{0\leq j\leq s} \omega \biggl(f^{(r+k)},{1\over n}\biggr)_j \log{n}\right] \] uniformly in \(x\) on \([-1,1]\) as \(n\rightarrow\infty\).