International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 3, Pages 153-158
doi:10.1155/S0161171203106151
On the convolution product of the distributional kernel Kα,β,γ,ν
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Received 18 June 2001; Revised 19 February 2002
Copyright © 2003 A. Kananthai and S. Suantai. 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
We introduce a distributional kernel Kα,β,γ,ν which is related to the operator ⊕ k iterated k
times and defined by ⊕ k=[(∑r=1p∂2/∂xr2)4−(∑j=p+1p+q∂2/∂xj2)4] k, where p+q=n is the dimension of the space ℝ n of the n-dimensional Euclidean space, x=(x1,x2,…,xn)∈ℝ n, k is a nonnegative integer, and α, β, γ, and ν are complex parameters. It is found that the existence of
the convolution Kα,β,γ,ν∗Kα′,β′,γ′,ν′ is depending on the conditions of p and q.