The Landau problem on compact intervals and optimal numerical differentiation
type: Item
instance of: scholarly article
title: The Landau problem on compact intervals and optimal numerical differentiation
zbMATH Open document ID: 0734.41009
DOI: 10.1016/0021-9045(90)90115-7
author: Henry Kallioniemi
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4214632
zbMATH Keywords: Landau problem
reviewed by: Peter Lappan
MaRDI profile type: Publication
full work available at URL: 0021-9045(90)90115-7
OpenAlex ID: W2068335746
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description: scientific article; zbMATH DE number 4214632