On the convolution equation related to the N-dimensional ultra-hyperbolic operator

We introduce the distribution eαt □k δ where □k is an ultra-hyperbolic operator iterated k times defined by □k ≡(∑i=1p∂2/∂ti2−∑j=p+1p+q∂2/∂tj2)k, k=0,1,2,…,p+q=n the dimension of the Euclidean space Rn. Now δ is the Dirac-delta distribution with □0δ=δ, □1δ=□δ and the variable t=(t1,t2,…,tn)∈Rn and the constant α=(α1,α2,…,αn)∈Rn with αt=α1t1+α2t2+⋯+αntn. First we study the property of eαt □k δ and after that we study its application of the convolution equation (eαt □k δ)∗u(t)=eαt∑r=0mCr □r δ where u(t) is the generalized function and Cr is a constant. The convolution equation is related to the ultra-hyperbolic equation. It is also found that the type of solutions of the convolution equation, such as the ordinary functions, the tempered distributions or the singular distributions depend on k,m and α.