International Journal of Mathematics and Mathematical Sciences
Volume 28 (2001), Issue 12, Pages 743-751

The sequential approach to the product of distribution

C. K. Li

Department of Mathematical Sciences, Augustana University College, Alberta, Camrose T4V 2R3, Canada

Received 12 June 2000

Copyright © 2001 C. K. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 rk 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 rk and Δlδ, where l1, by applying the normalization procedure due to Gel'fand and Shilov (1964). We begin this paper by distributionally normalizing Δrk 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 Δrkδ based on the results obtained by Li (2000), and Li and Fisher (1990).