Generalized Radon--Nikodym Spectral Approach. Application to Relaxation Dynamics Study

TL;DR

This paper introduces a Radon--Nikodym spectral method for analyzing relaxation dynamics, which constructs probability densities to extract dynamic characteristics without using traditional norm-based approaches, demonstrated on real and simulated signals.

Contribution

The paper develops a novel Radon--Nikodym spectral approach that avoids norm-based methods, enabling direct spectral analysis of relaxation signals and providing a new tool for relaxation dynamics study.

Findings

Successfully applied to model and experimental relaxation signals.

Provides a new way to estimate relaxation rate distributions.

Developed software implementation of the method.

Abstract

Radon--Nikodym approach to relaxation dynamics, where probability density is built first and then used to calculate observable dynamic characteristic is developed and applied to relaxation type signals study. In contrast with $L^2$ norm approaches, such as Fourier or least squares, this new approach does not use a norm, the problem is reduced to finding the spectrum of an operator (virtual Hamiltonian), which is built in a way that eigenvalues represent the dynamic characteristic of interest and eigenvectors represent probability density. The problems of interpolation (numerical estimation of Radon--Nikodym derivatives is developed) and obtaining the distribution of relaxation rates from sampled timeserie are considered. Application of the theory is demonstrated on a number of model and experimentally measured timeserie signals of degradation and relaxation processes. Software product, implementing the theory is developed.