Entropy and density approximation from Laplace transforms

How much information does the Laplace transforms on the real line carry about an unknown, absolutely continuous distribution? If we measure that information by the Boltzmann–Gibbs–Shannon entropy, the original question becomes: How to determine the information in a probability density from the given values of its Laplace transform. We prove that a reliable evaluation both of the entropy and density can be done by exploiting some theoretical results about entropy convergence, that involve only finitely many real values of the Laplace transform, without having to invert the Laplace transform.We provide a bound for the approximation error of in terms of the Kullback–Leibler distance and a method for calculating the density to arbitrary accuracy.