Weighted Fejér constants and Fekete sets (Q485548)

The weighted Lagrange interpolation polynomials on a system of nodes \(X=\{x_{k,n}:k=1,2,\dots ,n; n\in\mathbb{N}\}\) is \(L_{k,w,X}(x) =L_{k,w}(x)=w(x) \cdot\frac{\ell_{k}^{2}(x)}{w(x_{k})}\), where \(\ell_{k}(x)\), \(k=1,2,\dots ,n\), are the fundamental polynomials of Lagrange interpolation, and \(w(x)=e^{-Q(x)}\) is an approximating weight. If \(\| L_{k,w}\| _{\infty}=1\), then \((L_{k,w})'(x_{k})=0\) and the following constants \(C_{k},_{w}=\frac{\omega'' (x_{k})}{\omega'(x_{k})}+\frac{w'(x_{k}) }{w( x_{k}) }\) (Fejér constants) are null. A weighted Fekete set associated with a weight \(w\) on a set \(\Sigma \subset\mathbb{C}\) is the set \(\mathcal{F}_{n,w}=\{x_{1},\dots ,x_{n} \}\subset\Sigma\) such that the supremum \[ d_{n,w}=\sup_{z_{1},\dots,z_{n}\in\Sigma}d_{n,w}(z_{1},z_{2,}\dots, z_{n})=\sup_{z_{1},\dots ,z_{n}\in\Sigma}(\Pi| z_{i}-z_{j}| w( z_{i})w(z_{j}))^{\frac{2}{n(n-1)}} \] is attained at the set \(\mathcal{F}_{n,w}\). For the special weights \(w_{n}\) (defined by using \(w\) and the \(n\)th orthonormal polynomial \(p_{n,w}\) with respect to \(w\)), the Fekete set is formed by the zeros of \(p_{n,w}\). Let \(Y_{n,w}^{-}(x)=w(x)\sum_{k=1}^{n}\frac{\ell_{k}^{2} (x)}{w(x_{k})}= w(x)H_{n}\left( \frac{1}{w},(\frac{1}{w})';x\right)\) be the Hermite interpolation polynomial of \(\frac{1}{w}\) on the nodes \(\{x_{k}\}_{k=1}^{n}\). The main results of the paper may be summarized as it follows: If \(w\) is an admissible, approximating weight (cf. Def. 1) on \((a,b)\) and \(\left(\frac{1}{w}\right)^{(2n)}\geq0\) on \((a,b),\) then the following assertions are equivalent: (A) \(C_{k,w}=0,\) \(k=1,2,\dots ,n;\) (B) \(\mathcal{F}_{n,w^{\frac{1}{2(n-1)}}}=\{x_{1},x_{2},\dots ,x_{n}\}\); (C) \(Y_{n,w}(x)\) is \(w\)-stable and most economical (cf. Def. 6), i.e, it has minimal norm and minimum degree; (E) \(\{x_{1},x_{2},\dots ,x_{n}\}\) is \(1(w)\)-normal. Similar results hold for the special weight \(w_{n}\) satisfying the condition \(\left(\frac{1}{w_{n}}\right)^{(2n)}\geq0\) on \((a,b)\). Finally, the convergence of a weighted Grünwald interpolation process is proved.