On the use of Laplace transform inversion for reconstruction of relaxation spectra

We discuss the use of Fourier transforms to construct approximate inversion formulae for the Laplace transform and apply this technique to recover the relaxation spectrum from “data” for the relaxation modulus. We show that regularization by a Gaussian, as proposed by Davies and Anderssen [A.R. Davies, R.S. Anderssen, Sampling localization in determining the relaxation spectrum, J. Non-Newt. Fluid Mech. 73 (1997) 163–179] yields reasonable results even for data with significant noise. We also show that, in principle, other choices of regularization allow the relaxation spectrum to be reconstructed from data which are taken in any interval of time or frequency, however short and wherever located. We construct formulae which would do this, based on polynomial approximation of a function in an exponentially weighted space. This algorithm, however, turns out not to be practical, and we elucidate the reasons for that.