Polynomial approximation on touching domains in the complex plane (Q902188)

For \(\alpha>0\), consider the ``analytic continuation'' \(f_{\alpha}\) of \(|z|^{\alpha}\) defined by \[ f_{\alpha}(z)=\begin{cases} z^{\alpha}&\text{if }\text{Re}(z)>0,\\ (-z)^{\alpha} &\text{if }\text{Re}(z)<0, \\ 0 &\text{if }z=0.\end{cases} \] Suppose that \(G_1\) and \(G_2\) are open, bounded Jordan domains such that \([-1,0)\subset G_1\), \(\overline{G}_1\subset \{z\in\mathbb{C}: \text{Re}(z)<0 \}\cup \{0\}\) and \((0,1]\subset G_2\), \(\overline{G}_2\subset \{z\in\mathbb{C}: \text{Re}(z)>0 \}\cup \{0\}\), so that \(\overline{G}_1\cap\overline{G}_2=\{0\}\). Denote by \(K\) the closure of \(G_1\cup G_2\), and let \(E_n(f,K)\) be the best approximation of a function \(f: K\to \mathbb{C}\) by algebraic polynomials of order at most \(n\) in the space \(C(K)\). Let us say that the set \(K\) has the VP-property whenever \[ E_n(f_1,K)=O(n^{-1}) \quad \text{for}\quad n\to\infty. \] The author gives a full description of the family of all \(K\) with the VP-property. Moreover, the Gaier conjecture on the polynomial approximation of piecewise analytic functions [\textit{D. Gaier}, Complex Variables, Theory Appl. 34, No. 4, 325--342 (1997; Zbl 0923.30025)] is proved.