Generalized Function - Schwartz Distributions

Schwartz Distributions

The realization of such a concept that was to become accepted as definitive, for many purposes, was the theory of distributions, developed by Laurent Schwartz. It can be called a principled theory, based on duality theory for topological vector spaces. Its main rival, in applied mathematics, is to use sequences of smooth approximations (the 'James Lighthill' explanation), which is more ad hoc. This now enters the theory as mollifier theory.

This theory was very successful and is still widely used, but suffers from the main drawback that it allows only linear operations. In other words, distributions cannot be multiplied (except for very special cases): unlike most classical function spaces, they are not an algebra. For example it is not meaningful to square the Dirac delta function. Work of Schwartz from around 1954 showed that this was an intrinsic difficulty.

Some solutions to the multiplication problem have been proposed. One is based on a very simple and intuitive definition a generalized function given by Yu. V. Egorov (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.

Another solution of the multiplication problem is dictated by the path integral formulation of quantum mechanics. Since this is required to be equivalent to the Schrödinger theory of quantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions as shown by H. Kleinert and A. Chervyakov. The result is equivalent to what can be derived from dimensional regularization.