© Hindawi Publishing Corp. A CONVOLUTION PRODUCT OF (2j)th DERIVATIVE OF DIRAC’S DELTA IN r AND MULTIPLICATIVE DISTRIBUTIONAL PRODUCT BETWEEN r −k AND ∇( △ j δ)

The purpose of this paper is to obtain a relation between the 
distribution δ ( 2 j ) ( r ) and the operator △ j δ and to give a sense to the convolution distributional product 
 δ ( 2 j ) ( r ) ∗ δ ( 2 s ) ( r ) and the multiplicative 
distributional products r − k ⋅ ∇ ( △ j δ ) and ( r − c ) − k ⋅ ∇ ( △ j δ ) .

1. Introduction.Let x = (x 1 ,x 2 ,...,x n ) be a point of the n-dimensional Euclidean space R n .
We call ϕ(x) the C ∞ -functions with compact support defined from R n to R. Let and consider the functional r λ defined by (see [5, page 71]), where λ is a complex number and dx = dx 1 dx 2 •••dx n .For Re(λ) > −n, this integral converges and is an analytic function of λ.Analytic continuation to Re(λ) ≤ −n can be used to extend the definition of (r λ ,ϕ).
Calling Ω n to the hypersurface area of the unit sphere imbedded in the n-Euclidean space, we find in [5, page 71] that r λ ,ϕ = Ω n ∞ 0 r λ+n−1 S ϕ (r )dr , (1.3) where and dw is the hypersurface element of the unit sphere.
From [6, page 366, formula (3.4)], we know that the neutrix product of r −k and δ on R m exists and, furthermore, ) where k is a positive integer, m is the dimension of the space, j is the iterated Laplacian operator defined by (1.10), and ∇ is the operator defined by (1.9) In (1.7) and (1.8), by the symbol • we mean "neutrix product" which is defined by Li in [6, page 363, Definition 1.4, formula (1.11)].The purpose of this paper is to obtain a relation between the distribution δ (2j) (r ) and the operator j δ and to give a sense to convolution distributional product δ (2j) (r ) * δ (2s) (r ) and the multiplicative distributional products r −k • ∇( j δ) and (r − c) −k • ∇( j δ) which are showed in Sections 2, 3.1, 3.2, and 3.3.Here, j is defined by (1.10) and ∇ is the operator defined by (1.9).
We observed that relation (2.3) cannot be deduced from the formula δ (n+2j−1) (r ) = a j,n j δ (1.11) which appear in [1], where with n dimension of the space.

The relation between the distribution δ (2j
) (r ) and the operator j δ.In this section, we want to obtain a formula that relates the distribution δ (2j) (r ) to the operator j δ.

Applications of the basic formula (2.3).
In this section, we want to give a sense to the convolution distributional product of the form δ (2j) (r ) * δ (2s) (r ) and the distributional products r −k •∇( j δ) and (r − c) −k •∇( j δ).
We know from (2.3) that the following formula is true: From (3.1), δ (2j) (r ) is a finite linear combination of δ and its derivatives, in consequence, we conclude that δ (2j) (r ) is a distribution of the class O c , where O c [7, page 244] is the space of rapidly decreasing distributions.Therefore, using the formula [2, page 75, formula (26)], where t is the iterated Laplacian operator defined by (1.10), we obtain the following formula: where In particular, letting j = s = 0 in (3.3), we have where

The multiplicative distributional product of r −k • ∇( j δ).
To give a sense to the multiplicative distributional product of we must study the cases r −2k • ∇( j δ) and r 1−2k • ∇( j δ) where ∇ is the operator defined by (1.9) and j is the iterated Laplacian operator defined by (1.10).

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