On the recovery and resolution of exponential relaxational rates from experimental data: Laplace transform inversions in weighted spaces

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Abstract
In three previous papers the authors have considered the problem of Laplace transform inversion when the unknown function is of bounded, strictly positive support. The improvement in resolution due to a priori knowledge of the support was quantified and methods for the choice of an optimum sampling of data were given. The authors discuss some undesirable edge effects encountered in practice with these methods and indicate a way to refine such calculations by considering the problem of Laplace transform inversion in weighted L2 spaces. A smoothly varying weight takes into account a partial knowledge of the localisation of the solution which can be estimated a priori from the knowledge of its first and second moments which are easily derived from the data before inversion. In such a way the reconstructed solution is forced to be small where it is likely to be small and the troublesome edge effects found in the previous methods are suppressed. The results show somewhat surprising improvements in typical inversions. The extensions of the method required for the analysis of sampled and truncated experimental data are also discussed and applied.