Delta-Shaped Kernels Generated by Laplace Transform Inversion Methods

The application of the integral Laplace transform for a wide class of problems leads to a simpler equation for the image of the desired original. At the next step, the inversion problem, i.e., finding the original from its image, arises. As a rule, it is not possible to carry out this step analytically. The problem of using approximate inversion methods arises. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points of the complex half-plane, in which the image is regular. However, unlike the image, the original may even have break points. For most inversion methods, there are no error estimates, which makes it difficult to compare methods with each other and to choose a specific method for practical application. The goal of this paper was to consider various methods from a unified point of view, namely, to study the so-called delta-shaped kernels generated by them, as well as issues of constructing delta methods with specified properties, estimating their errors, accelerating convergence, etc. The ideology of delta methods apparently belongs to Widder, although in an implicit form.