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 N greater then a constant N(k, y_i) that depends only on k N and $_i=1,...,s-1{ y_i - y_i+1}, 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: ((y_i ρ_n(y_i), y_i + ρ_n(y_i)),ρ_n(x) := 1/n2 + 1 - x^2/n, P_n(y_i) = 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.