Pointwise estimation of an almost copositive approximation of continuous functions by algebraic polynomials

  • H. A. Dzyubenko

Abstract

In the case where a function continuous on a segment $f$ changes its sign at $s$ points $y_i : 1 < y_s < y_{s-1} < ... < y_1 < 1$, for any $n \in N$ greater then a constant $N(k, y_i)$ that depends only on $k \in N$ and \$\min_{i=1,...,s-1}\{ y_i - y_{i+1}\}$, we determine an algebraic polynomial $P_n$ of degree \leq n such that: $P_n$ has the same sign as f everywhere except possibly small neighborhoods of the points $y_i$: ($$(y_i \rho_n(y_i), y_i + \rho_n(y_i)),\quad \rho_n(x) := 1/n2 + \sqrt{1 - x^2}/n,$$ $P_n(y_i) = 0$ and $$| f(x) P_n(x)| \leq c(k, s)\omega_k(f, \rho_n(x)),\quad x \in [ 1, 1],$$ where $c(k, s)$ is a constant that depends only on $k$ and $s$ and $\omega k(f, \cdot )$ is the modulus of continuity of the function $f$ of order $k$.
Published
25.05.2017
How to Cite
DzyubenkoH. A. “Pointwise Estimation of an Almost Copositive Approximation of continuous functions by Algebraic Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 5, May 2017, pp. 641-9, http://umj.imath.kiev.ua/index.php/umj/article/view/1723.
Section
Research articles