International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 6, Pages 361-369
doi:10.1155/S0161171200004233
The product of r−k and ∇δ on
ℝm
University College of Cape Breton, Mathematics and Computer Science, Nova Scotia, Sydney B1P 6L2, Canada
Received 21 December 1999; Revised 4 January 2000
Copyright © 2000 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.
Abstract
In the theory of distributions, there is a general lack of
definitions for products and powers of distributions. In physics
(Gasiorowicz (1967), page 141), one finds the need to evaluate
δ2 when calculating the transition rates of certain
particle interactions and using some products such as
(1/x)⋅δ. In 1990, Li and Fisher introduced a
“computable” delta sequence in an m-dimensional space to obtain
a noncommutative neutrix product of r−k and Δδ (Δ denotes the Laplacian) for any
positive integer k between 1 and m−1 inclusive. Cheng and
Li (1991) utilized a net δϵ(x) (similar to the
δn(x)) and the normalization procedure of μ(x)x+λ to deduce a commutative neutrix product of
r−k and δ for any positive real number k. The object
of this paper is to apply Pizetti's formula and the normalization
procedure to derive the product of r−k and ∇δ (∇ is the gradient operator) on ℝm. The nice properties
of the δ-sequence are fully shown and used in the proof of
our theorem.