Pointwise estimation of an almost copositive approximation of continuous functions by algebraic polynomials - Dzyubenko, H. A. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
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 determinean algebraic polynomial $P_n$ of degree ≤ n such that: $P_n$ has the same sign as f everywhere except possibly small neighborhoods of the points $y_i$:($$(y_i ρ_n(y_i), y_i + ρ_n(y_i)),\quad ρ_n(x) := 1/n2 + \sqrt{1 - x^2}/n,$$$P_n(y_i) = 0$ and$$| f(x) P_n(x)| ≤ c(k, s)ω_k(f, ρ_n(x)),\quad x \in [ 1, 1],$$where $c(k, s)$ is a constant that depends only on $k$ and $s$ and $ω k(f, · )$ is the modulus of continuity of the function $f$ oforder $k$.