Harmonic version of Jackson's theorem in the complex plane (Q1365560)

Suppose that \(L\) is a Jordan arc in the complex plane and that \(f\) is continuous on \(L\), with modulus of continuity \(\omega_f\). If \(L\) is a straight line segment, then for each \(n=1,2,3, \dots,\) there exists a polynomial \(p_n\) in the complex variable \(z\), of degree at most \(n\), such that for all \(z\in L\), \(|p_n(z) -f(z) |\leq c\omega_f (1/n)\), for some positive constant \(c\). This is the classical Jackson theorem. D. J. Newman asked whether this result (which we will call property \((J))\) held for less restricted \(L\), and conjectured that \((J)\) would hold if \(L\) were smooth. On the other hand, Newman stated that he could not prove that \((J)\) fails for \(L\) consisting of 2 line segments meeting at a right angle. It was later shown by Andersson that \((J)\) does fail for an \(L\) shaped \(L\) and by Lesley that \((J)\) fails for smooth \(L\). Work of Mamedhanov and of Maimeskul gives weak sufficient conditions for \((J)\) to hold. The paper under review considers the same question for approximation of real valued continuous functions by harmonic polynomials, and the corresponding approximability property is called \((JH)\). It is shown that if \(L\) is piecewise \(C^{2,\alpha}\) smooth (for example, an \(L\) shaped \(L)\) then the property \((JH)\) does hold. Thus the two properties \((J)\) and \((JH)\) are essentially different.