On Local Stability in the Complete Prony Problem

Abstract

We consider the variational Prony problem on approximating observations \(x \) by the sum of exponentials. We find critical points and the second derivatives of the implicit function \(\theta \) that relates perturbation in \(x \) with the corresponding exponents. We suggest upper bounds for the second order increments and describe the domain, where the accuracy of a linear approximation of \(\theta \) is acceptable. We deduce lower estimates of the norm of deviation of \(\theta \) for small perturbations in \(x \). We compare our estimates of this norm with upper bounds obtained with the use of Wilkinson’s inequality.