A modulus of smoothness based on an algebraic addition
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.
scientific article; zbMATH DE number 1053599