Numerical recovery of location and residue of poles of meromorphic functions

TL;DR

This paper introduces a novel numerical method to recover the location and residue of poles of meromorphic functions from finite, noisy samples on the positive real axis, without relying on rational approximation or regularization techniques.

Contribution

It presents a stable, convergent algorithm that estimates pole parameters directly from sampled data, avoiding traditional rational approximation and regularization methods.

Findings

Method is numerically stable and converges with increasing data points.

Can evaluate the approximation error of pole estimates from noisy data.

Does not require rational function approximation or regularization.

Abstract

We present a method able to recover location and residue of poles of functions meromorphic in a half--plane from samples of the function on the real positive semi-axis. The function is assumed to satisfy appropriate asymptotic conditions including, in particular, that required by Carlson's theorem. The peculiar features of the present procedure are: (i) it does not make use of the approximation of meromorphic functions by rational functions; (ii) it does not use the standard methods of regularization of ill-posed problems. The data required for the determination of the pole parameters (i.e., location and residue) are the approximate values of the meromorphic function on a finite set of equidistant points on the real positive semi-axis. We show that this method is numerically stable by proving that the algorithm is convergent as the number of data points tends to infinity and the noise on the input data goes to zero. Moreover, we can also evaluate the degree of approximation of the estimates of pole location and residue which we obtain from the knowledge of a finite number of noisy samples.