Reconstructing Stieltjes functions from their approximate values: a search for a needle in a haystack

TL;DR

This paper presents a novel algorithm for reconstructing Stieltjes functions from discrete measurements, providing optimality certificates and uncertainty visualization, with applications demonstrated in electrochemical impedance spectroscopy.

Contribution

A new reconstruction algorithm for Stieltjes functions that guarantees optimality and visualizes uncertainty, advancing analysis of material response functions.

Findings

Algorithm successfully reconstructs Stieltjes functions from limited data.

Provides certificates of optimality and uncertainty visualization.

Effective in electrochemical impedance spectroscopy applications.

Abstract

Material response of real, passive, linear, time-invariant media to external influences is described by complex analytic functions of frequency that can always be written in terms of Stieltjes functions -- a special class of analytic functions mapping complex upper half-plane into itself. Reconstructing such functions from their experimentally measured values at specific frequencies is one of the central problems that we address in this paper. A definitive reconstruction algorithm that produces a certificate of optimality as well as a graphical representation of the uncertainty of reconstruction is proposed. Its effectiveness is demonstrated in the context of the electrochemical impedance spectroscopy.