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

  • 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 $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ 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
How to Cite
Kopotun, K. A., D. Leviatan, and I. A. Shevchuk. “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.
Section
Research articles