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I am wondering if the function $\mathrm{sinc}(x)=\frac{\sin x}{x}$ can be represented in terms of Fourier series?

Thank you.

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It's not periodic, so you'll need to restrict the domain somewhat. If the restricted domain is symmetric about the origin, your series becomes a cosine series since the sine cardinal is even. –  J. M. Dec 17 '11 at 1:52
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Were you possibly thinking about the Fourier transform of the sinc function which turns out to be a rect function ? –  Dilip Sarwate Dec 17 '11 at 3:09
    
No, I am thinking about integral $\int_{-\pi}^{\pi}|sinc{x}|^ndx$. –  David Dec 17 '11 at 4:16
    
I did not understand about the cosing sing. Could you elaborate, please. thank you! –  David Dec 17 '11 at 4:17

1 Answer 1

Well, in order to compute the Fourier series of any function, you need to specify the interval $(-\ell,\ell)$. But once you do, sure you can find the Fourier series of $f(x)={\sin x\over x}$ in the usual way.

For example, taking $\ell=\pi$, here is a plot of $f(x)$ (black) and the first three terms of its Fourier series (blue):

Mathematica graphics

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