Find the Fourier transform of $$\dfrac{\sin(a t)}{t\pi}$$
I tried using the formula but I can't get it to work, I asked my teacher and he said to use the proprietary of duality, but I don't understand how. Could someone please explain how? Or give the solution?
I think you have to use the cardinal sine function somewhere. I also tried dividing the functions in smaller parts but that brought me nowhere. I really don't know how to approach this.
Following Dave's hint, $$\int_{-a/2\pi}^{a/2\pi}\exp 2\pi iktdk=\frac{\sin at}{\pi t}.$$By the inversion theorem,$$\int_{\Bbb R}\frac{\sin at}{\pi t}\exp -2\pi ikt dt=\chi_{[-a/2\pi,\,a/2\pi]}(k).$$This is one definition of the Fourier transform of $\frac{\sin at}{\pi t}$.