Inversion from different kinds of information and real inversion formulas of the Laplace transform from a finite number of data

Abstract

We propose a new method to obtain solutions (and generalized solutions) of general bounded linear operator equations with a parameter on Hilbert spaces. This will be done by using reproducing kernel Hilbert spaces, and some of their properties. As a result of the method, we shall propose a principle to obtain an optimal and natural inverse from only a finite number of given data. In addition, as a prototype example, we shall give new real inversion formulas of the Laplace transform.