On the divergence of polynomial interpolation in the complex plane (Q5945738)

The author extends the results of H.-P. Blatt, M. Götz, L. Brutman and I. Gopengauz from the divergence of Hermite-Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, he proves the following theorem: Let \(\Delta_n\subset\mathbb{C}\); \(\Delta_n=\left\{z_1^{(n)}, z_2^{(n)},\dots, z_n^{(n)}\right\}\). Let \(\varphi_1^{(n)}, \varphi_2^{(n)},\dots, \varphi_n^{(n)}\) be polynomials (of arbitrary degree) such that \(\varphi_j^{(n)}(z_k^{(n)})=\delta_{jk}\). Let \(z_0\in\mathbb{C}\) be an accumulation point of \(\{\Delta_n\}\). Then for any neighbourhood \(U\ni z_0\); \(U\subset\mathbb{C}\): \(\sup_{z\in U}\Lambda_n(z)\to\infty\) as \(n\to\infty\). Here \(\Lambda_n(z)=\sum_{k=1}^n|\varphi_k^{(n)}(z)|\).