Optimal Lebesgue function type sums
Let \(X_n\) be the set of all grids \(\xi=(\xi_1,\xi_2,\dots,\xi_n)\) with \(1\leq\xi_1<\xi_2<\dots<\xi_n\leq-1\). Denote by \(\ell_k(\xi, \cdot)\) the fundamental functions of Lagrange interpolation on the grid \(\xi\in X_n\). Then the Lebesgue function type sum \[ S_n(\xi,x):=\sum^n_{k=1}\left| \left(x-\xi_k\right) \ell_k \left(\xi,x\right)\right| ^s\quad (x\in [-1,1]) \] with fixed \(s>0\) plays a fundamental role in the mean convergence of Lagrange and Hermite-Fejér interpolation. The author considers the interesting problem to find an optimal grid \(\xi^*\in X_n\) with \[ \left\| S_n \left(\xi^*, \cdot \right)\right\| _\infty = \inf\left\{\left\| S_n\left(\xi, \cdot \right)\right\| _\infty; \xi \in X_n\right\}. \] He has proved that for fixed \(s>0\) the optimal grid is uniquely given by the Chebyshev nodes \(\xi^*_k=\cos\frac{(2k-1)\pi}{2n}\).
mean convergence of Lagrange interpolation
scientific article; zbMATH DE number 1244253