Pointwise Estimation of Sign-Preserving Polynomial Approximations on Arcs in the Complex Plane

In 2014, V. Andrievskii proved that if a real-valued function f ∈ Lip 𝛼, 0 < 𝛼 < 1, defined on a given smooth Jordan curve satisfying the Dini condition changes its sign finitely many times, then it can be approximated by a harmonic polynomial that changes its sign on the indicated curve at the same points as f, and the approximation error has the same order as the classical Dzyadyk’s error of pointwise approximation. By applying the scheme of the proof proposed by Andrievskii, we generalize his result to the case of an arbitrary modulus of continuity 𝜔(f, t) satisfying the inequality γ𝜔(f, 2t) ≥ 𝜔(f, t), where γ = const < 1.