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A new proof of a theorem concerning the set \(\{ N^-_k (x)\}^n_{k=0}\). (English) Zbl 0917.41019
The set \(\{N_k^-(x)\}_{k=0}^n\) is defined on the complex plane \(\mathbb{C}\) with \(N_k^-(x)=| x |^k\), if \(k\) is even and \(N_k^-(x)=x| x |^{k-1}\), if \(k\) is odd. Let \(V^-_n(x_1,\dots,x_n) \) denote the \(n\times n\) matrix with the \((i,j)\) entry \(N^-_{i-1}(x_j)\). The author earlier [Approximation Theory Appl. 12, No. 2, 45-53 (1996; Zbl 0861.41025)] discussed some approximation properties of the functions \(N^-_k(x)\). Among others he proved the following theorem: “If \(x_1,\dots,x_n \in \mathbb{C}\) are distinct and no three of them have the same modulus, then \[ \det V^-_n(x_1,\dots,x_n)=\prod_{1\leq i<j\leq n}(a_{ji}x_j-a_{ji}x_i)\not =0, \] where \(a_{ji}, a_{ij} \in \mathbb{C}\) such that \(| a_{ij}| =| a_{ji}|=1\) for all \(1\leq i < j \leq n\).” In the paper under review the author gives a new proof to this theorem.
Reviewer: M.Lenard (Kuwait)
41A50 Best approximation, Chebyshev systems