The Landau problem on compact intervals and optimal numerical differentiation (Q810783)

Let \(0<m<n\), and let f be either a real or complex-valued function on a compact interval I such that f is n-times differentiable on I and \(f^{(n)}\) is bounded on I. The Landau problem in the title is to obtain a bound on \(f^{(m)}\) on I when the bounds on f and \(f^{(n)}\) are known. The author obtains an estimate on \(f^{(m)}\) by obtaining an estimate on the nth derivative of the Lagrange polynomial L which interpolates f at n points of I, where these n-points are chosen to facilitate the estimate on both \(L^{(n)}\) and the ``error term'' for \(\| L^{(n)}-f^{(n)}\|_{\infty}\). The resulting bound called for in the Landau problem is approximately \(4^ me\) times smaller than an earlier bound due to \textit{H. Cartan} [Sur les classes des inégalités portant sur leurs decrivées successives, Actualiés Sci. Ind. No.867 (Paris, 1940)].