International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 16, Pages 833-845
doi:10.1155/S0161171204305193
On defining the product r−k⋅∇lδ
Department of Mathematics and Computer Science, Brandon University, Brandon R7A 6A9, Manitoba, Canada
Received 15 May 2003
Copyright © 2004 C. K. Li and V. Zou. 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
Let ρ(s) be a fixed infinitely differentiable function defined on R+=[0,∞) having the properties: (i) ρ(s)≥0, (ii) ρ(s)=0 for s≥1, and (iii) ∫Rmδn(x)dx=1 where δn(x)=cmnmρ(n2r2) and cm is the constant satisfying (iii). We overcome difficulties arising from computing ∇lδn and express this regular sequence by two mutual recursions and use a Java swing program to evaluate corresponding coefficients. Hence, we are able to imply the distributional product r−k⋅∇lδ for k=1,2,… and l=0,1,2,… with the help of Pizetti's formula and the normalization.