A modulus of smoothness based on an algebraic addition
type: Item
instance of: scholarly article
title: A modulus of smoothness based on an algebraic addition
zbMATH Open document ID: 0882.41008
DOI: 10.1007/BF02755446
author: Michael Felten
published in: Aequationes Mathematicae
publication date: 1998-02-15T00:00:00Z
full work available at URL: https://eudml.org/doc/137694
review text: The Bernstein-Jackson theorems relate the smoothness of a \(2\pi\)-periodic function \(f\) to the rapidity of convergence to \(f\) of, \(t_n\), the best approximation to \(f\) from the trigonometric polynomials of degree \(n\) or less. This paper defines a new modulus of continuity, \(w(f)\), for the function \(f\) defined on \([-1,1]\) so that a similarly elegant statement can be made for convergence by algebraic polynomials. Although the definition of \(w\) is too cumbersome for a short review such as this, the following statement illustrates the economy of the definition once it is defined. Theorem: \((E_nf)_x =O(n^{-\alpha}) \Leftrightarrow w(f,\delta)_x = O(\delta^{\alpha}).\) \noindent Here \(E_nf\) represents the distance from \(f\) to the polynomials of degree less than or equal \(n\). \noindent This paper is related to the work of \textit{Z. Ditzian} and \textit{V. Totik} [``Moduli of smoothness'' (1987; Zbl 0666.41001)] who also defined a new modulus of continuity for this purpose. Besides proving the Bernstein-Jackson theorem in this setting, this paper develops the properties of the new modulous of continuity and demonstrates some clear advantages over the Ditzian-Totik modulus.
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 1053599
zbMATH Keywords: Bernstein theorem
reviewed by: Daniel Wulbert
MaRDI profile type: Publication
cites work: Chebyshev transform methods in the theory of best algebraic approximation
cites work: Q3817079
cites work: Q4855449
cites work: Characterization of best algebraic approximation by an algebraic modulus of smoothness
cites work: Q5835448
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description: scientific article; zbMATH DE number 1053599