On a special interpolation problem and overconvergence (Q1874153)

A (polynomial) interpolation problem is said to be regular when the interpolant is unique. Let \(f=\sum c_{k}z^{k}\) be a function analytic in the open disk \(D_{\rho }=\left\{ z:\left|z\right|<\rho \right\} \) and not on the closed disk. For \(l\geq 1\) let \(S_{nl-1}(z)\) be the partial sum of the series. Let \( p_{1,n}(f,z) (p_{1,n}(S_{nl-1},z))\) be the \((0,1)\)-interpolating polynomial to \(f\) (to \(S_{nl-1})\) of degree at most \(2n-1,\) where the nodes are the zeros of \(z(z^{n}-\alpha ^{n})\) \((0<\alpha <\rho).\) The author proves that \[ \lim_{n\rightarrow \infty }(p_{1,n}(f,x)-p_{1,m}(S_{nl-1},z))=0, \] for \(\left|z\right|<\rho (\rho /\alpha)^{(l/2)-1}\).