Lagrange interpolation polynomials in \(E^ p (D)\) with \(1<p< +\infty\)
This paper deals with the approximation problem by the Lagrange interpolation polynomials for the complex functions which belong to a generalized Hardy space \(E^p(D)\) with \(1<p<\infty\), where \(D\) is a Jordan domain with boundary \(\Gamma\) which is assumed to be piecewise \(C^2\) smooth with no cusps in the paper. For a function \(f\in E^p(D)\) its norm \(|f|_p\) is defined as the \(L^p\) norm of its non-tangential boundary value on \(\Gamma\). The author constructed appreciate interpolation nodes \(z_{k,n}\), \(k=0,1,\dots, n-1\), \(n\in\mathbb{N}\), and then proved \[ \lim_{n\to\infty}|f-L_{n-1}|_p=0,\qquad \forall f\in E^p(D), \] where \(L_{n-1}(f)\) denotes the \(n-1\)-th Lagrange interpolation polynomial to \(f\) based on the nodes \(z_{k,n}\), \(k=0,1,\dots, n-1\).
scientific article; zbMATH DE number 878086