IJPAM: Volume 62, No. 2 (2010)

ON THE SOLUTION OF n-DIMENSIONAL OPERATOR
k AND THE FOURIER TRANSFORM OF
THEIR CONVOLUTION

Wanchak Satsanit
Department of Mathematics
Faculty of Science
Maejo University
Nong Han, San Sai, Chiang Mai, 50290, THAILAND
e-mail: aunphue@live.com


Abstract.In this paper, we consider the solution of equation

\begin{displaymath}\circledast^{k}u(x)=\sum^{m}_{r=0}c_{r}\circledast^{r}\delta\end{displaymath}

where $\circledast^{k}$ is the circledast operator iterated $k$ times and is defined by

\begin{eqnarray*} \circledast^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2... ...j=p+1}\frac{\partial^2}{\partial x^2_j}\right)^{3}\right)^{k}, \end{eqnarray*}

$x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$ - the $n$-dimensional Euclidian space, $p+q=n$, $c_{r}$ is a constant, $\delta$ is the Dirac-delta distribution, $\circledast^{0}\delta=\delta$ and $k=0,1,2,3,\cdots$. It was found that the type of the solution of this equation, such as the ordinary functions, the tempered distributions and the singular distributions, depend on the relationship between the values of $k$ and $m$. After that we study the Fourier transform of the elementary solution and also the Fourier transform of their convolution.

Received: April 27, 2010

AMS Subject Classification: 46F10, 46F12

Key Words and Phrases: ultra-hyperbolic kernel, diamond operator, tempered distribution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 62
Issue: 2