ON THE SPECTRUM OF THE DISTRIBUTIONAL KERNEL RELATED TO THE RESIDUE

We study the spectrum of the distributional kernel Kα,β(x), where α and β are complex numbers and x is a point in the space Rn of the n-dimensional Euclidean space. We found that for any nonzero point ξ that belongs to such a spectrum, there exists the residue of the Fourier transform (−1)k ̂ K2k,2k(ξ), where α = β = 2k, k is a nonnegative integer and ξ ∈Rn. 2000 Mathematics Subject Classification. 46F10, 46F12.


Introduction. Gel'fand and Shilov
have studied the generalized function P λ , where is a quadratic form, λ is a complex number, and p + q = n is the dimension of R n .They found that P λ has two sets of singularities, namely λ = −1, −2,...,−k,... and λ = −n/2, −n/2 − 1,...,−n/2 − k,..., where k is a positive integer.For the singular point λ = −k, the generalized function P λ has a simple pole with residue for p + q = n is odd with p odd and q even.Also, for the singular point λ = −n/2 − k they obtained res λ=−n/2−k P λ = (−1) q/2 L k δ(x) for p + q = n is odd with p odd and q even.Now, let K α,β (x) be the convolution of the functions R H α (u) and R β (v), that is, where R H α (u) and R β (v) are defined by (2.1) and (2.3), respectively.Since R H α (u) and R β (v) are tempered distributions, see [4, pages 30-31], thus K α,β (x) is also a tempered distribution and is called the distributional kernel.
In this paper, we use the idea of Gel'fand and Shilov to find the residue of the Fourier transform (−1) k K 2k,2k (ξ), where K 2k,2k is defined by (1.4) with α = β = 2k and k is a nonnegative integer.We found that for any nonzero point ξ that belongs to the spectrum of (−1) k K 2k,2k (x), there exists the residue of the Fourier transform where δ is the Dirac-delta distribution.The operator k was first introduced by Kananthai [4] and named as the Diamond operator defined by where p + q = n is the dimension of R n .Moreover, the operator k can be expressed as the product of the operators k and k , that is, where k is an ultra-hyperbolic operator iterated k times defined by where p + q = n.The operator k is an elliptic operator or Laplacian iterated k times defined by Trione [7, page 11] has shown that the function R H 2k (u) defined by (2.1) with α = 2k is an elementary solution of the operator k .Also, Aguirre Téllez [1, pages 147-148] has proved that the solution R H 2k (u) exists only for odd n with p odd and q even (p + q = n).Moreover, we can show that the function (−1) k R 2k (v) is an elementary solution of the operator k , where R 2k (v) is defined by (2.3) with β = 2k.

Preliminaries
Definition 2.1.Let x = (x 1 ,x 2 ,...,x n ) be a point of R n , and write u of an interior of the forward cone, and Γ + denotes the closure of Γ + .For any complex number α, define where the constant K n (α) is given by the formula 3) The function R β (v) is called the elliptic kernel of Marcel Riesz and is an ordinary function for Re(β) ≥ n and is a distribution of β for Re(β) < n.Definition 2.3.Let f be a continuous function, then the Fourier transform of f , denoted by f or f (ξ), is defined by ) If f is a distribution with compact support, by [8, Theorem 7.4.3,page 187] (2.5) can be written as (2.6) Lemma 2.4.Given the equation where k is the operator defined by (1.5), and δ is the Dirac-delta distribution, u(x) is an unknown, k is a nonnegative integer and x ∈ R n , where n is odd with p odd, q even In this paper, we study the spectrum of (−1) k K 2k,2k (x), relate to the residue of the Fourier transform (−1) k K 2k,2k (ξ).
Definition 2.6.The spectrum of the distributional kernel K α,β (x) is the support of the Fourier transform K α,β (ξ) or the spectrum of K α,β (x) = supp K α,β (ξ).Now, from Lemma 2.5 we obtain (2.10) In particular, from (2.9) the spectrum of (2.11) Lemma 2.7.Let P (x 1 ,x 2 ,...,x n ) be a quadratic form of positive definite, and is defined by (2.12) then for any testing function ϕ(x) ∈ D, the space of infinitely differentiable function with compact support, where where dΩ p and dΩ q are the elements of surface area on the unit sphere in R p and R q , respectively.Both integrals (2.13) and (2.14 , these integrals must be understood in the sense of their regularization and (2.13) defined as δ  4) du, (2.16) where ψ 1 (u, v) = ψ(r , s).
where K α,β is defined by (1.4) and x, ξ ∈ R n , then K α,β (x) can be extended to the entire function K α,β (z) and be analytic for all z = (z 1 ,z 2 ,...,z n ) ∈ C n , where C n is the n-tuple space of complex number and where exp(b| Proof.Since the integral of (2.18) converges for all ξ ∈ G b , thus K α,β (x) can be extended to the entire function K α,β (z) and be analytic for all z ∈ C n .Thus (2.18) can be written as where We must show that the support of K α,β (ξ) is contained in G b .Since K α,β (z) is an analytic function that satisfies the inequality (2.19) and is called an entire function of order of growth ≤ 1 and of type ≤ b, then by Paley-Wiener-Schartz theorem, see [3, page 162], K α,β (ξ) has a support contained in G b , that is the spectrum of In particular, for α = β = 2k, the spectrum of (−1) k K 2k,2k (x) is also contained in G b , that is supp[(−1) k K 2k,2k (ξ)] ⊂ G b , where (−1) k K 2k,2k (x) is an elementary solution of the Diamond operator k by Lemma 2.4, and the Fourier transform (−1) k K 2k,2k (ξ) given by (2.9) can be defined as follows.