Reconstructing an Analytic Function Using Truncated Lagrange Polynomials

Abstract

Let $U$ be the unit disc of the complex plane. We consider the problem of reconstructing a function $f$ in the Hardy space $H^2(U)$ from values $f(z_n^{(m)})$, where $\{z_n^{(m)}\}$, $(m\in \mathbb{N}$; $1\le n\le m)$, is a given point system in $U$. This is an ill-posed problem. The function $f$ is approximated by so-called truncated Lagrange polynomials. Necessary and sufficient conditions for the convergence are established and a regularization result is given.