Fourier series of $\operatorname{sinc}(x)$

Question

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

Thank you.

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):