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

Authors

  • H. A. Dzyubenko

Abstract

In the case where a function continuous on a segment f changes its sign at s points yi:1<ys<ys1<...<y1<1, for any nN greater then a constant N(k,yi) that depends only on kN and \mini=1,...,s1{yiyi+1}, we determine an algebraic polynomial Pn of degree \leq n such that: Pn has the same sign as f everywhere except possibly small neighborhoods of the points yi: ((yiρn(yi),yi+ρn(yi)),ρn(x):=1/n2+1x2/n, Pn(yi)=0 and |f(x)Pn(x)|c(k,s)ωk(f,ρn(x)),x[1,1], where c(k,s) is a constant that depends only on k and s and ωk(f,) is the modulus of continuity of the function f of order k.

Published

25.05.2017

Issue

Section

Research articles

How to Cite

Dzyubenko, H. 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, https://umj.imath.kiev.ua/index.php/umj/article/view/1723.