Distribution of interpolation points
Let \(E\) be a compact subset of the complex plane with regular boundary \(\partial E\) and with connected complement. Let \(f: E\to\mathbb{C}\) be a continuous function such that there exists a sequence of polynomials of degree at most \(n\) with \(\varlimsup_{n\to\infty}|f-P_n|^{1/n}_E={1\over r}\) for \(r>1\) and such that \(f-P_n\) has \(n+1\) zeros on \(\partial E\). The author proves that the sequence of counting measures associated to these points contains a subsequence which converges to the equilibrium measure on \(E\) in the weak sense. Conversely, the analyticity of a function \(f\) on an open set containing \(E\) is proved, provided that the sequence of measures mentioned above converges to a unit measure \(\sigma\) and additionally the interpolation polynomials tend to \(f\) geometrically.
scientific article; zbMATH DE number 896435