(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find a distribution [tex]g_n[/tex] which satisfies

[tex] g'_n(x) = \delta(x - n) - \delta(x + n) [/tex]

and use it to prove

[tex] \lim_{n \to \infty} \frac{\sin{nx}}{\pi x} = \delta(x) [/tex]

2. Relevant equations

Nothing relevant comes up at the moment.

3. The attempt at a solution

Well the first part is pretty easy I think. The distribution would be

[tex] g_n(x) = \theta(x - n) - \theta(x + n) = \left\{ \begin{array}{l l}

-1 & \quad |x| < n \\

0 & \quad |x| \geq n \\

\end{array} \right.[/tex]

The limit will indeed resemble a delta function when [tex]n[/tex] goes to infinity and π is probably just a normalization constant. But applying the two Heaviside functions to solve this has got me stumped.

P.S. Gotta catch some sleep, I will be back in 7 hours hopefully with some ideas to solve this.

Cheers

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# Generalized functions (distributions) problem - Mathematical physics

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