The sequential approach to the product of distribution

It is well known that the sequential approach is one of the main tools of dealing with product, power, and convolution of distribution (cf. Chen (1981), Colombeau (1985), Jones (1973), and Rosinger (1987)). Antosik, Mikusiński, and Sikorski in 1972 introduced a definition for a product of distributions using a delta sequence. However, δ2 as a product of δ with itself was shown not to exist (see Antosik, Mikusiński, and Sikorski (1973)). Later, Koh and Li (1992) chose a fixed δ-sequence without compact support and used the concept of neutrix limit of van der Corput to define δk and (δ′)k for some values of k. To extend such an approach from one-dimensional space to m-dimensional, Li and Fisher (1990) constructed a delta sequence, which is infinitely differentiable with respect to x1,x2,…,xm and r, to deduce a non-commutative neutrix product of r−k and Δδ. Li (1999) also provided a modified δ-sequence and defined a new distribution (dk/drk)δ(x), which is used to compute the more general product of r−k and Δlδ, where l≥1, by applying the normalization procedure due to Gel'fand and Shilov (1964). We begin this paper by distributionally normalizing Δr−k with the help of distribution x+−n. Then we utilize several nice properties of the δ-sequence by Li and Fisher (1990) and an identity of δ distribution to derive the product Δr−k⋅δ based on the results obtained by Li (2000), and Li and Fisher (1990)