Pointwise Estimation of the Almost Copositive Approximation of Continuous Functions by Algebraic Polynomials

In the case where a function f continuous on a segment changes its sign at s points yi : − 1 < ys< ys−1< ... < y1< 1, for any n ∈ ℕ greater than a certain constant N(k, yi) that depends only on k ∈ ℕ and mini=1,…,s−1yi−yi+1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \underset{i=1,\dots, s-1}{\min}\left\{{y}_i-{y}_{i+1}\right\}, $$\end{document} we determine an algebraic polynomial Pn of degree ≤ n such that: Pn has the same sign as f everywhere except, possibly, small neighborhoods of the points yi:yi−ρnyiyi+ρnyi,ρnx≔1/n2+1−x2/n,Pnyi=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {y}_i:\left({y}_i-{\rho}_n\left({y}_i\right),{y}_i+{\rho}_n\left({y}_i\right)\right),\kern1em {\rho}_n(x):= 1/{n}^2+\sqrt{1-{x}^2}/n,\kern1em {P}_n\left({y}_i\right)=0, $$\end{document} 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.