IJPAM: Volume 71, No. 1 (2011)

ON THE CONVOLUTION EQUATION
RELATED TO THE KLEIN-GORDON OPERATOR

Amphon Liangprom$^1$, Kamsing Nonlaopon$^2$
Department of Mathematics
Khon Kaen University
Khon Kaen, 40002, THAILAND


Abstract. In this paper, we study the distribution $e^{\alpha x} (\square +m^2)^{k} \delta$, where $(\square +m^2)^{k}$ is the Klein-Gordon operator iterated $k$ times defined by ([*]), $k$ is a non-negative integer, $\delta$ is the Dirac-delta distribution, $m$ is a non-negative real number, $x=(x_{1}, x_{2}, \ldots ,x_{n})$ is a variable and $\alpha = (\alpha_{1}, \alpha_{2}, \ldots ,\alpha_{n}) $ is a constant and both are the points in the $n$-dimensional Euclidean spaces $\mathbb{R}^{n}$.

At first, the properties of $e^{\alpha x} (\square +m^2)^{k} \delta$ are studied and after that we study the application of $e^{\alpha x} (\square +m^2)^{k} \delta$ for solving the solution of the convolution equation

\begin{displaymath}e^{\alpha x} (\square +m^2)^{k} \delta\ast u(x)=e^{\alpha x}\sum_{r=0}^M C_r(\square +m^2)^{r}\delta,\end{displaymath}

where $u(x)$ is the generalized function and $C_r $ is a constant. It found that the type of solutions of this convolution equation, such as the ordinary function and the singular distribution depend on the relationship between the values of $k$ and $M$.

Received: May 5, 2011

AMS Subject Classification: 46F10, 46F12

Key Words and Phrases: convolution equation, tempered distribution, Klein-Gordon operator, Dirac-delta distribution

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 71
Issue: 1