A generalization of Lagrange interpolation theorem (Q1082551)

If f is an analytic function defined on the closed unit disc \(\bar D,\{Z_ i\}\) is a sequence of distinct points in \(\bar D\) such that lim \(Z_ n=0\) and \(P_{n-1}\) is the Lagrange interpolation polynomial of f at \(Z_ 1,Z_ 2,...,Z_ n\), it is known that \(P_{n-1}\) converges to f in the sup norm on \(\bar D.\) That is, f(Z) can be recovered as a limit from the given sequence \(f(Z_ k).\) The author considers a similar setting and asks the question of whether one can reconstruct f(Z) if one is given the data \(f(Z_ i)+e_ i\) where the \(\{e_ i\}\) are assumed to be independent identically distributed random vectors with mean zero and finite variance. The author employs a minimization scheme to verify this assertion.