I'm currently struggling with this:
Calculate the fourier transform of $t^2·δ(t-a)$
I've tried applying the derivative property of the fourier transform to simplify things but to no avail. Would there be a way of applying the sifting property to make things easier? I also tried an inverse tranform of the convultion of the two functions but things became way to messy. I think there has to be a simpler solution that I'm missing.
It's easier than you think: $$\mathcal F\{ t^2 \delta(t-a) \} = \int_{-\infty}^{\infty} t^2 \delta(t-a) e^{-i \omega t} \, dt = \left. t^2 e^{-i \omega t} \right|_{t=a} = a^2 e^{-i \omega a}$$ where it has been used that $\int_{-\infty}^{\infty} f(t) \, \delta(t-a) \, dt = f(a).$