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 y _i : − 1 < y _s < y _s−1 < ... < y ₁ < 1, for any n ∈ ℕ greater than a certain constant N(k, y _i ) that depends only on k ∈ ℕ and $\underset{i=1,\dots, s-1}{\min}\left\{{y}_i-{y}_{i+1}\right\},$ we determine an 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 ${y}_i:\left({y}_i-{\rho}_n\left({y}_i\right),{y}_i+{\rho}_n\left({y}_i\right)\right),\kern1em {\rho}_n(x):= 1/{n}^2+\sqrt{1-{x}^2}/n,\kern1em {P}_n\left({y}_i\right)=0,$ and |f(x) − P _n (x)| ≤ c(k, s)ω _k (f, ρ _n (x)), x ∈ [−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 of order k.