On an extremal problem of Fejér
Let \(-\infty \leq a<b\leq \infty\) and let \({\mathcal P}_{n+1}\) denote the class of \((n+1)\)-triples \((x_ 1,x_ 2,...,x_{n+1})\) with \(a\leq x_ 1<...<x_{n+1}\leq b\) \((x_ 1\) and \(x_{n+1}\) are finite). Let w(x) be a non-negative function on [a,b] and let \(\ell_ i(x)\) denote the fundamental polynomials of Lagrange on the \((n+1)\)-triples. The authors consider the extremal problem of finding the value \[ N=\inf_{P_{n+1}}\sup_{a\leq x\leq b}\{w(x)\sum^{n+1}_{j=1}\ell^ 2_ j(x)/w(x_ j)\}=\inf_{P_{n+1}}\tilde N(x_ 1,...,x_{n+1}) \] when w(x) is a non-negative function on [a,b]. They also determine the set of points \(\{x_ j\}_ 1^{n+1}\) which minimizes \(\tilde N\) for a fixed w(x). This problem when \(w(x)=1\) goes back to Fejer. For other relevant results see the book on Chebyshev systems by S. Karlin and W. J. Studden and a paper by \textit{C. Balazs} [Acta Math. Acad. Sci. Hungar. 34, 307-315 (1979; Zbl 0434.41004)].
scientific article; zbMATH DE number 4084243