On the convolution equation related to the diamond kernel of Marcel Riesz

In this paper, we study the distribution eαt◊kδ where ◊k is introduced and named as the Diamond operator iterated k-times (k = 0,1,2,…) and is definded by ◊k = ((α2αt12 + α2αt22 + … + α2αtp2)2 - (α2αtp+12 + α2αtp+22 + … + α2αtp+q2)2)k, where t = (t1,t2,…,tn) is a variable and α = (α1,α2,…,αn) is a constant and both are the points in the n-dimensional Euclidean space Rn, δ is the Dirac-delta distribution with ◊0δ = δ and p + q = n (the dimension of Rn) st, the properties of eαt◊kδ are studied and later we study the application of eαt◊kδ for solving the solutions of the convolution equation (eα1◊kδ)∗u(t) = eα1∑r=0m cr◊rδ. We found that its solutions related to the Diamond Kernel of Marcel Riesz and moreover, the type of solutions such as, the classical solution (the ordinary function) or the tempered distributions depending on m, k and α.