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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)
##### MSC:
 41A50 Best approximation, Chebyshev systems