On interpolation theory in the complex domain

D. L. Berman

Izvestiya. Mathematics , Tome 6 (1972) no. 4, pp. 782-787

Résumé

It is shown that for the nodes $z_k^{(n)}=e^{i\theta_k^{(n)}}$, where $\theta_k^{(n)}=\frac{(2k+1)\pi}n$, $k=1,\dots,n$; $n=1,2,\dots$, the following statements hold: 1) The Hermite–Fejér interpolation process for an arbitrary polynomial converges in $|z|\leqslant1$ with rapidity $O\bigl(\frac1n\bigr)$. 2) The process $R_n(f,z)=\sum_{k=1}^nf\bigl(z_k^{(n)}\bigl)\bigl[l_k^{(n)}(z)\bigr]^2$, where $\bigl\{l_k^{(n)}(z)\bigr\}$ are Lagrange fundamental polynomials with nodes $\bigl\{z_k^{(n)}\bigr\}$, diverges at all points $z\ne0$ of $|z|\leqslant1$ for every function $f(z)=z^s$, $s=0,1,2,\dots$ .