Divergence of interpolation polynomials in the complex domain

Abstract:

In 1918 L. Fejér gave an example of a function $f(z)$, analytic for $|z| < 1$ and continuous for $|z| \leqq 1$, such that the sequence of Lagrange polynomials found by interpolation to $f(z)$ at the roots of unity diverges at a point on the unit circle. More recently S. Ja. Al’per showed that, regardless of how the interpolation points are chosen on the unit circle, a function $g(z)$, analytic for $|z| < 1$ and continuous for $|z| \leqq 1$, exists such that the Lagrange polynomials do not converge uniformly to $g(z)$ for $|z| \leqq 1$. In the present paper we present some theory which sheds some light on the results of Fejér and Al’per. A new example of the divergence of Lagrange polynomials is also presented.