Pointwise estimation of sign-preserving polynomial approximation on arcs in the complex plane

Abstract

UDC 517.53

V. Andrievskii proved in 2014 that if a real-valued function $f\in{\rm Lip}\,\alpha,$ $0<\alpha <1,$ 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 curve at the same points as $f$ and the order of approximation error is the same as the classical Dzyadyk error of pointwise approximation. Applying the Andrievskii proof scheme, we generalize this result to the case of an arbitrary modulus of continuity $ \omega (f, t)$ under the condition $\gamma\omega (f, 2t) \geq \omega (f, t) ,$ where $ \gamma = {\rm const} <1.$