On local stability in the complete Prony problem

In the variational Prony problem of approximating observations $x$ by the sum of exponentials, expressions are obtained for critical points and second derivatives of the implicit function $\theta(x)$ of exponents' dependence on $x$ data. Upper bounds are proposed for the second order increments $\|\Delta_{2}\theta\|$ with a description of the $\Delta x$ area where $\theta(x)$ is approximately linear. As a consequence, the lower bounds for $\|\Delta\theta\|$ are obtained for small perturbations in $x$. A comparison with the upper bounds for $\|\Delta\theta\|$ by Wilkinson's inequality is given.