Reconstructing an analytic function using truncated Lagrange polynomials (Q1432987)

Let \(\{z_{m,n}\}\), \(m=1,2,\ldots,n\), \(n=1,2,\ldots\) be a system of points in the unit disk \(D\), such that for each \(m\) the points \(z_{m,1}, z_{m,2}, \ldots, z_{m,m}\) are distinct. Given a function \(f\in H^2(D)\) put \(\mu_{m,n}=f(z_{m,n})\) and \(\nu=(\mu_{m,1},\ldots,\mu_{m,m})\). The convergence of standard Lagrange polynomials \(L_m(\nu)\) to the function \(f\) uniformly in \(\overline D\) is governed by the Kalmar-Walsh theorem. The authors study the similar problem in \(H^2\) by using the truncated Lagrange polynomials. They prove that such convergence holds under much less restrictive assumptions on the system \(z_{m,n}\) and so the Kalmar-Walsh theorem is not true if \(C(\bar D)\) is replaced by \(H^2\).