Real inversion formulas and numerical experiments of the Laplace transform by using the theory of reproducing kernels

We shall give very natural and numerical real inversion formulas of the Laplace transform for the general data following the ideas of best approximations, generalized inverses, the Tikhonov regularization and the theory of reproducing kernels. However, the new method in this paper for the real inversion formulas of the Laplace transform is based essentially on a Fredholm integral equation of the second kind. These real inversion formulas may be expected to be practical to calculate the inverses of the Laplace transform by computers when the real data contain noises or errors. We shall illustrate examples, by using computers.