A characteristic of points in \({\mathbb{R}}^ 2\) having Lebesgue function of polynomial growth
Let \(E=\{x_ 1,x_ 2,...,x_ N\}\) be \(N:=\binom{n+m}{m}\) points in \(C^ m\) such that the Vandermondian \(V_ n(x_ 1,...,x_ n)\neq 0\), where V is the determinant \(M_ n\) of the monomials. If \(V_ n\neq 0\) let the Lebesgue function of the interpolating problem be denoted by \(\Lambda_ n(x):=\sum^{n}_{i=1}| \ell_ i(x)|.\) The problem of characterizing the set E of points for which \(\Lambda_ n(x)\) is minimum is very difficult. Here the author considers the case when \(m=2\) and E is the unit disk in \({\mathbb{R}}^ 2\). The author shows that in this case, for the set of points in E for which the Lebesgue function has polynomial growth, we have \(\lim_{n\to \infty}V_ n^{3/n(n+1)(n+2)}=(2e)^{-1/2}.\) In one variable, the analogue of this limit is known to be \(1/2\) when \(E=[-1,1]\). [\textit{G. M. Goluzin}, Geometric Theory of Functions of a Complex Variable (1969; Zbl 0148.306)].
scientific article; zbMATH DE number 4105507