On complex valued functions with strongly unique best Chebyshev approximation (Q2366700)

Let \(f\) be a continuous function defined on a compact subset \(Q\) of the complex plane \(\mathbb{C}\), \(C(Q)\) denotes the space of continuous complex valued functions on \(Q\), and \(V\) an \(n\)-dimensional Haar subspace of \(C(Q)\). An element \(v\in V\) is called a best Chebyshev approximation to \(f\) if \(\| f-v\|= \min_{w\in V} \| f-w\|\). The best approximation \(v\) to \(f\) is called strongly unique if there is a real number \(r>0\) such that for each \(w\in V\), \(\| f-w\|\geq \| f- v\|+ r\| v-w\|\). In contrast to the complex case, the best Chebyshev approximation with respect to a finite dimensional Haar subspace \(V\subset C(Q)\) is always strongly unique if all functions are real valued. However, strong uniqueness still holds for complex valued function \(f\) with a so-called reference of maximal element. \textit{H. P. Blatt} [ibid. 41, 159-169 (1984; Zbl 0537.41015)] proved that this class forms on open and dense subset in \(C(Q)\) if the number of isolated points of \(Q\) does not exceed \(\dim V\). In this paper, it is shown that this result also holds in the space \(A(Q)\) of functions, analytic in the interior of \(Q\), if the compact set \(Q\) satisfies the regularity condition: The complement \(\mathbb{C}| Q\) is connected and if to each point \(z_ 0\) of the boundary \(\partial Q\) of \(Q\) there exists a continuous function \(\varphi: Q|\{z_ 0\}\to \mathbb{R}\) and a constraint \(r>0\) such that \(\varphi(Q| \{z_ 0\})\subset [-r,+r]\) and \(z-z_ 0= (z-z_ 0)e^{i\varphi(z)}\) for \(z\in Q| \{z_ 0\}\).