Optimal Extrapolation of the Derivative of a Polynomial Approximately Defined on the Lemniscate

We study the problem of optimal extrapolation (optimal recovery) of the derivative of a polynomial defined with an error on a compact set in the complex plane when the compact set is a lemniscate. Along with the extrapolation problem, we consider a related sharp inequality for derivatives of polynomials. An exact solution to the optimal extrapolation problem is obtained when the compact set is a connected lemniscate and the extrapolation point lies on its exterior. For an arbitrary lemniscate, we construct a nearly optimal extrapolation method.