# Calculate the nth Fourier transform of the Fejer kernel, i.e., $N\int\Big(\frac{\sin \pi xN}{\pi xN}\Big)^2 e^{-2\pi ixn}dx$

I want to calculate the Fourier transform of the Fejer kernel $\mathcal{F}_N$ on the real line, which is given by $$\mathcal{F}_N(x)= \begin{cases} N\Big(\frac{\sin \pi xN}{\pi xN}\Big)^2, & \text{if x\neq 0} \\[2ex] N, & \text{if x=0} \end{cases}$$

I want to find $\hat{\mathcal{F}_N} (n)=N\int\Big(\frac{\sin \pi xN}{\pi xN}\Big)^2 e^{-2\pi ixn}dx$, but so far I can't think of any transformations to compute this integral. I would greatly appreciate any help.

• You can use the convolution theorem $$\int_{-\infty}^{\infty} dx \ f(x) g^*(x) e^{i k x} = \frac1{2 \pi} \int_{-\infty}^{\infty} dk' \ F(k') G^*(k-k')$$ where $f$ and $F$ are Fourier transform pairs, as are $g$ and $G$. In this case both $f$ and $g$ are $\sin{\pi N x}/{\pi N x}$ so that $F$ and $G$ are $\pi N \operatorname{rect}{\pi N k}$. – Ron Gordon Oct 8 '16 at 11:34
• @RonGordon I'm not familiar with the notation rect here can you explain what it means? – takecare Oct 8 '16 at 11:39
• It is a rectangular pulse of width $1/(\pi N)$ centered at the origin. – Ron Gordon Oct 8 '16 at 11:40
• Is it possible to calculate the Fourier transform of $\sin x/x$ via straightforward integration or do I need to use a different trick? – takecare Oct 8 '16 at 11:41
• Oh yes, it is possible. I would do it via contour integration in the complex plane. – Ron Gordon Oct 8 '16 at 11:41