Tuning Fork Delta Sequence/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

The graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approaches $-\infty$. Only taking the entire graph we can ensure the proper behaviour for $n$ approaching $\infty$,

Let $\sequence {\map {\delta_n} x}$ be a sequence such that:

$\map {\delta_n} x := \begin{cases}

-n & : \size x < \frac 1 {2n} \\ 2n & : \frac 1 {2n} \le \size x \le \frac 1 n \\ 0 & : \size x > \frac 1 n \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:

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x\) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {\int_{- \infty}^{-\frac 1 n } \map \phi x \map {\delta_n} x \rd x + \int_{- \frac 1 n }^{- \frac 1 {2n} } \map \phi x \map {\delta_n} x \rd x + \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \map \phi x \map {\delta_n} x \rd x + \int_{\frac 1 {2n} }^{\frac 1 n} \map \phi x \map {\delta_n} x \rd x + \int_{\frac 1 n}^{\infty} \map \phi x \map {\delta_n} x \rd x}\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \map \phi x \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \map \phi x \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \map \phi x \rd x}\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} n \paren {2 \map \phi {\xi_1} \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \map \phi {\xi_2} \paren {\frac 1 {2n} - \paren {- \frac 1 {2n} } } + 2 \map \phi {\xi_3} \paren {\frac 1 n - \frac 1 {2n} } }\) Mean Value Theorem for Integrals, $\xi_1 \in \closedint {-\frac 1 n } {-\frac 1 {2n} }$, $\xi_2 \in \closedint {-\frac 1 {2n} } {\frac 1 {2n} }$, $\xi_3 \in \closedint {\frac 1 {2n} } {\frac 1 n }$
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren { \map \phi {\xi_1} - \map \phi {\xi_2} + \map \phi {\xi_3} }\)
\(\ds \) \(=\) \(\ds \map \phi {\lim_{n \mathop \to \infty} \xi_1} - \map \phi {\lim_{n \mathop \to \infty} \xi_2} + \map \phi {\lim_{n \mathop \to \infty} \xi_3}\) Limit of Image of Sequence on Real Numbers
\(\ds \) \(=\) \(\ds \map \phi 0 - \map \phi 0 + \map \phi 0\) $\ds \lim_{n \mathop \to \infty} \frac 1 n = 0$, Squeeze Theorem for Real Sequences
\(\ds \) \(=\) \(\ds \map \delta \phi\) Definition of Dirac Delta Distribution

Furthermore:

\(\ds \int_{-\infty}^\infty \map {\delta_n} x \rd x\) \(=\) \(\ds 2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \rd x\)
\(\ds \) \(=\) \(\ds n \paren {2 \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \paren {\frac 1 {2n} - \paren {- \frac 1 {2n} } } + 2 \paren {\frac 1 n - \frac 1 {2n} } }\)
\(\ds \) \(=\) \(\ds n \paren {\frac 2 {2n} - \frac 2 {2n} + \frac 2 {2n} }\)
\(\ds \) \(=\) \(\ds 1\)

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Tuning_Fork_Delta_Sequence/Proof_1&oldid=617665"