IJPAM: Volume 68, No. 1 (2011)

ON THE FOURIER TRANSFORM OF
THE DIAMOND KLEIN-GORDON KERNEL

Apisit Lunnaree$^1$, Kamsing Nonlaopon$^2$Department of Mathematics, Khon Kaen University, Khon Kaen, 40002, THAILAND
$^{1,2}$Department of Mathematics
Khon Kaen University
Khon Kaen, 40002, THAILAND
$^1$e-mail: 5250200709@stdmail.kku.ac.th
$^2$e-mail: nkamsi@kku.ac.th
$^2$Centre of Excellence in Mathematics
CHE, Si Ayutthaya Rd., Bangkok 10400, THAILAND


Abstract. In this article, the operator $(\diamondsuit+m^2)^k$ is introduced and named as the diamond Klein-Gordon operator iterated $k$-times and is defined by

\begin{displaymath}(\diamondsuit+m^2)^k=\left[\left(\sum_{i=1}^p\frac{\partial^2... ...}^{p+q}\frac{\partial^2}{\partial x_j^2}\right)^2+m^2\right]^k,\end{displaymath}

where $p+q=n$ is the dimension of the space $\mathbb{R}^n$, for $x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n,$ $m$ is a nonnegative real number and $k$ is a nonnegative integer. In this work, we study the fundamental solution of operator $(\diamondsuit+m^2)^k$ and this fundamental solution is called the diamond Klein-Gordon kernel. Then, we study the Fourier transform of the diamond Klein-Gordon kernel and also the Fourier transform of their convolution.

Received: January 18, 2011

AMS Subject Classification: 46F10

Key Words and Phrases: Dirac-delta distribution, Fourier transform, tempered distribution, diamond Klein-Gordon kernel

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 68
Issue: 1