The product of r−k and ∇δ on ℝm - C. K. Li (International Journal of Mathematics and Mathematical Sciences)
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.