Rectangular Delta Sequence
Theorem
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
- $\map {\delta_n} x := \begin{cases}
0 & : x < - \frac 1 {2n} \\ n & : - \frac 1 {2n} \le x \le \frac 1 {2n} \\ 0 & : x > \frac 1 {2n} \end{cases}$
Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence.
That is, in the distributional sense it holds that:
- $\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$
or
- $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x = \map \delta \phi$
where $\phi \in \map \DD \R$ is a test function, $\delta$ is the Dirac delta distribution, and $\map \delta x$ is the abuse of notation, usually interpreted as an infinitely thin and tall spike with its area equal to $1$.
Proof
Let $\phi \in \map \DD \R$ be a test function.
Then:
| \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \map \phi x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} n \map \phi {\xi_n} \paren {\frac 1 {2n} - \paren {- \frac 1 {2n} } }\) | Mean Value Theorem for Integrals, $\xi_n \in \closedint {-\frac 1 {2n} } {\frac 1 {2n} }$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map \phi {\xi_n}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \phi {\lim_{n \mathop \to \infty} \xi_n}\) | Limit of Image of Sequence on Real Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \phi 0\) | $\ds \lim_{n \mathop \to \infty} \frac 1 {2n} = \lim_{n \mathop \to \infty} \paren {- \frac 1 {2n} } = 0$, Squeeze Theorem for Real Sequences | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \delta \phi\) | Definition of Dirac Delta Distribution |
$\blacksquare$
Sources
2013: George Arfken, Hans J. Weber and Frank E. Harris: Mathematical Methods for Physicists (7th ed.): Chapter $1$ Mathematical Preliminaries $1.11$ Dirac Delta Function