IJPAM: Volume 35, No. 3 (2007)

CAUSAL (ANTICAUSAL) CONVOLUTION PRODUCTS OF
THE DISTRIBUTIONAL FAMILIES RELATED TO
THE ULTRAHYPERBOLIC KLEIN GORDON OPERATOR,
ULTRAHYPERBOLIC OPERATOR, LAPLACIAN
OPERATOR AND THE DIAMOND OPERATOR

Manuel A. Aguirre T.
Núcleo Consolidado Matemática Pura y
Aplicada (NuCOMPA)
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Pinto 399, Tandil, 7000, ARGENTINA
e-mail: maguirre@exa.unicen.edu.ar

Abstract.Let $E_{\alpha, \beta}$ and $T_{\alpha, \beta}$ the convolution distributional functions families defined by $E_{\alpha, \beta}=G_{\alpha}\ast H_{\beta}$ and $T_{\alpha, \beta}=G_{\alpha}\ast R_{\beta},$ where $G_{\alpha}=G_{\alpha}(P\pm i0, m, n)$ is the causal (anticausal) distribution defined by ()(cf. [#!T1!#]) and $H_{\beta}=H_{\beta}(P\pm i0, n)$ is causal (anticausal) analogues of the elliptic kernel of M.Riesz (cf. [#!T1!#]) defined by () and $R_{\beta}$ is the elliptic kernel of Marcel Riesz defined by (). In this paper we give a sense to convolution product of $E_{\alpha, \beta}\ast E_{\alpha^{\lq }, \beta^{\lq }}$ and $T_{\alpha, \beta}\ast T_{\alpha^{\lq }, \beta^{\lq }}$ for all $\alpha, \beta, \alpha^{\lq }, \beta^{\lq }$ complex numbers such that $\beta, \beta^{\lq }$ and $\beta+\beta^{\lq }\neq n+2r, r=0, 1, \dots$ . As consequence of our formula we give a sense to convolution product of: $K^{k}\delta\ast K^{k^{\lq }}\delta, L^{k}\delta\ast L^{k^{\lq }}\delta, \triangle ^{k}\delta\ast\triangle^{k^{\lq }}\delta$ and $\Diamond^{k}\delta\ast \Diamond^{k^{\lq }}\delta$ , where $K^{k}$ is the -dimensions ultrahyperbolic Klein Gordon operator iterated -times, $L^{k}$ is the ultrahyperbolic operator iterated -times, $\triangle^{k}$ is the Laplacian operator iterated -times and $\Diamond^{k}$ is the Diamond operator iterated -times.