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.

  • $\begingroup$ I do know that the Fourier transform of the $[-1/2,1/2]$ indicator function is $$\frac{\sin(\pi\xi)}{\pi\xi}.$$ $\endgroup$ – Dave Mar 23 at 15:16
  • $\begingroup$ Could you double-check your title and body edits? I'm sure you meant $\frac{\sin at}{\pi t}$ (\frac{\sin at}{\pi t}). $\endgroup$ – J.G. Mar 23 at 15:31
  • $\begingroup$ Do you mean ${\sin(at)\over t\pi}?$ $\endgroup$ – saulspatz Mar 23 at 15:33
  • $\begingroup$ math.stackexchange.com/questions/839378/… $\endgroup$ – Aditya Garg Mar 23 at 15:34

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 $\frac{\sin at}{\pi t}$.


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