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Find Fourier transform of $t f^{2}(t)$
$\mathcal{F}(\omega)=\int_{-\infty}^{\infty}\left\{t f^{2}(t)\right\} e^{j \omega t} d t$
Using time multiplication of FT: $t^{n} f(t)=(j)^{n} \frac{d^{n}}{d \omega^{n}} F(\omega)$
assuming $g(t)=f^{2}(t)$
Therefore, $\mathcal{F}(\omega)=\int_{-\infty}^{\infty}\{\operatorname{tg}(t)\} e^{j \omega t} d t=j \frac{d}{d \omega^{n}} G(\omega)$ since $g(t)=f^{2}(t)$
using another FT property: $f^{n}(t)=j^{n} \omega^{n} F(\omega)$
Therefore, $G(\omega)=j^{2} \omega^{2} F(\omega)$
Finally, $\mathscr{F}(\omega)=\int_{-\infty}^{\infty}\left\{t f^{2}(t)\right\} e^{j \omega t} d t=j \frac{d}{d \omega^{n}} j^{2} \omega^{2} F(\omega)$