EXPONENTIAL-SAMPLING METHOD FOR LAPLACE AND OTHER DILATIONALLY INVARIANT TRANSFORMS .1. SINGULAR-SYSTEM ANALYSIS

In this paper we show that the exponential-sampling method, introduced by Ostrowsky et al for the fast numerical inversion of the Laplace and similar transforms, by analogy with the well known Nyquist sampling method in the theory of Fourier transformation, can also be used for the approximate computation of the singular systems of these operators when a priori knowledge of the (finite) support of the solution is available. The resulting approximation in a singular-function series has a very interesting feature (which also holds in the Fourier case): since the solution is a linear combination of sampling functions it is (Mellin) band-limited but it is also approximately space-limited, the spatial localization being provided by a standard 'profile' function related to the interpolation and decay of the sampling functions within and at the edges of the support, respectively. We also discuss both the more practical case of data integrated over finite steps between sampling points and a 'zoom' technique for closer inspection of regions of interest in the reconstruction.