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 problem of inversion arises, i. e. finding the original from its image. As a rule, it is not possible to carry out this step analytically. The problem arises of using approximate inversion methods. 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, the original, unlike the image, may even have break points. For most handling methods there are no error estimates, which makes it difficult to compare methods with each other and the choice of a particular method for practical application. Target of this work is to consider various methods from a single point of view, namely the study of the so-called called delta-shaped nuclei, as well as design issues delta methods with specified properties, estimating their error, accelerating convergence, etc. Ideology of delta methods belongs, apparently, to Widder, although in an implicit form.