On the recovery and resolution of exponential relaxation rates from experimental data. - III. The effect of sampling and truncation of data on the Laplace transform inversion

Abstract

The method of singular function expansions, which in previous papers in this series was used for the inversion of the Laplace transform for the cases, respectively, of continuous data and continuous solution and discrete data and discrete solution, is extended to cover the case of discrete data and continuous solution. Two discrete data points distributions are considered: uniform and geometric. For both we prove that the singular values and the singular functions (in the solution space) of the problem with discrete data and continuous solution converge to the singular values and singular functions of the problem with continuous data and continuous solution when the number of points tends to infinity and the distance between adjacents points tends to zero. Furthermore, we show by means of numerical computations that for geometrically sampled data it is possible to obtain even better approximations of the greatest singular values than in the discrete-to-discrete case using a similar number of data points. Excellent approximations of the continuous-to-continuous case singular functions are also obtained. Implementation of the inversion procedure gives continuous solutions with high computational efficiency.