Sinc^2 as a delta function representation?

[Loro]

Hi, it's actually not homework but a part of my research.

I intuitively see that:

$\lim_{t \rightarrow \infty} \frac{sin^2[(x-a)t]}{(x-a)^2} \propto \delta(x-a)$

I know it's certainly true of $sinc$, but I couldn't find any information about $sinc^2$. Could someone give me a hint on how I could prove it, and find the proportionality coefficient?

 

[Dick]

Hi, it's actually not homework but a part of my research.

I intuitively see that:

$\lim_{t \rightarrow \infty} \frac{sin^2[(x-a)t]}{(x-a)^2} \propto \delta(x-a)$

I know it's certainly true of $sinc$, but I couldn't find any information about $sinc^2$. Could someone give me a hint on how I could prove it, and find the proportionality coefficient?

A representation of the delta function should have integral 1. Put a=0 and integrate it. You'll find it needs more than just a constant normalization.

 

[Loro]

Thanks, so it seems:

$\lim_{t \rightarrow \infty} \frac{1}{\pi t} \frac{sin^2[(x-a)t]}{(x-a)^2}$

does what I want.

 

[Dick]

Thanks, so it seems:

$\lim_{t \rightarrow \infty} \frac{1}{\pi t} \frac{sin^2[(x-a)t]}{(x-a)^2}$

does what I want.

Sounds right.