Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Authors

  • K. A. Kopotun Univ. Manitoba, Winnipeg, Canada
  • D. Leviatan Tel Aviv Univ., Israel
  • I. A. Shevchuk

Abstract

In Part I of the paper, we have proved that, for every α>0 and a continuous function f, which is either convex (s=0) or changes convexity at a finite collection Ys={yi}si=1 of points yi(1,1), sup where E_n (f) and E^{(2)}_n (f, Y_s) denote, respectively, the degrees of the best unconstrained and (co)convex approximations and c(α, s) is a constant depending only on α and s. Moreover, it has been shown that N^{∗} may be chosen to be 1 for s = 0 or s = 1, α ≠ 4, and that it must depend on Y_s and α for s = 1, α = 4 or s ≥ 2. In Part II of the paper, we show that a more general inequality \sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\}, is valid, where, depending on the triple (α,N,s) the number N^{∗} may depend on α,N,Y_s, and f or be independent of these parameters.

Published

25.03.2010

Issue

Section

Research articles

How to Cite

Kopotun, K. A., et al. “Are the Degrees of the Best (co)convex and Unconstrained Polynomial Approximations the Same? II”. Ukrains’kyi Matematychnyi Zhurnal, vol. 62, no. 3, Mar. 2010, pp. 369–386, https://umj.imath.kiev.ua/index.php/umj/article/view/2873.