IJPAM: Volume 72, No. 1 (2011)

An asymptotic product for $X^s\delta^{(k)}(r^2 - t^2)$

C.K. Li
Department of Mathematics and Computer Science
Brandon University
Brandon, Manitoba, R7A 6A9, CANADA
[email protected]


Abstract. For any Schwartz testing function $\phi$, the distribution $\delta^{(k)}(r^2 - t^2)$ focused on the sphere $O_t$ of $r = t$ in $R^n$ is defined by

\begin{displaymath}
(\delta^{(k)}(r^2 - t^2), \, \phi) = \frac{(-1)^k}{2 t^{n -...
...c{\partial}{2 r \partial r}\right)^k
(\phi r^{n - 2}) d O_t,
\end{displaymath}

which is the solution of the wave equation with the initial conditions described below in a space of odd dimension:

\begin{eqnarray*}
&& (\triangle - \frac{\partial^2}{\partial t^2}) u = 0 \\
&...
...\partial u(x, 0)}{\partial t} = (-1)^k 2 \pi^{k + 1}\delta (x).
\end{eqnarray*}


We apply the well-known Pizzetti's formula

\begin{eqnarray*}
S_\phi(r) & \sim & \phi(0) + \frac{1}{2!}S''_\phi(0)r^2 + \cd...
...k \phi(0)r^{2 k}}{2^k \,
k! \, n (n + 2) \cdots (n + 2 k - 2)}
\end{eqnarray*}


to derive an asymptotic expansion for the distribution $\delta^{(k)}(r^2 - t^2)$ and obtain an asymptotic product for $X^s \, \delta^{(k)}(r^2 - t^2)$ based on the formula

\begin{displaymath}
\triangle^k (\phi \psi) = \sum_{m + i + l = k} 2^i\binom{m ...
...l}\nabla^i \triangle^m \phi \cdot \nabla^i
\triangle^l \psi.
\end{displaymath}

This product should have potential applications in seeking certain solutions for the differential equations involving the gradient operator $\nabla$ in distributional sense.

Received: July 1, 2011

AMS Subject Classification: 46F10

Key Words and Phrases: distribution, product, asymptotic expansion, asymptotic product, wave equation and Pizzetti's formula

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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: 72
Issue: 1