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

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 $k$.