About integrating product of two sinc function using Fourier transform So the problem is 
which I think is pretty straight-foward by using Fourier transform and convolution property of two sinc functions and evaluating the convolution at 5. However, I got sinc(t) for the convolution result(So the answer is sinc(5)?). But when I use matlab to check, it says that the result is 100sinc(t)(again 100sinc(5))?  I'm really confused and got stuck here.
 A: The convolution theorem states that a convolution of two functions is equal to the inverse FT of the product of the FTs of each of the functions.  That is,
$$\int_{-\infty}^{\infty} dx' \, f(x') g(x-x') = \frac1{2 \pi} \int_{-\infty}^{\infty} dk \, F(k) G(k) e^{-i k x}$$
where
$$F(k) = \int_{-\infty}^{\infty} dx \, f(x) e^{i k x}$$
$$G(k) = \int_{-\infty}^{\infty} dx \, g(x) e^{i k x}$$
So when each of $f$ and $g$ are $\operatorname{sinc}{x} = \sin{x}/x$, then $F(k) = G(k) = \pi 1_{[-1,1]} $.
Then
$$\int_{-\infty}^{\infty} dx' \operatorname{sinc}{x'} \operatorname{sinc}{(x-x')} = \frac{\pi}{2 \pi} \int_{-1}^1 dk \,  e^{i k x} = \operatorname{sinc}(x)$$
So, yes, the answer to your question is $\operatorname{sinc}(5)$.  No idea how Matlab could return what it did.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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$\ds{\bbox[5px,#ffd]{}}$

\begin{align}
&\bbox[5px,#ffd]{%
\int_{-\infty}^{\infty}\on{sinc}\pars{t}\on{sinc}\pars{t - 5}\,
\dd t}
\\[5mm] = &\
\int_{-\infty}^{\infty}
\bracks{{1 \over 2}\int_{-1}^{1}\expo{\ic\omega t}\dd\omega}
\bracks{{1 \over 2}\int_{-1}^{1}
\expo{-\ic\nu\pars{t - 5}}\,\,\dd\nu}\dd t
\\[5mm] = &\
{\pi \over 2}\int_{-1}^{1}\expo{5\nu\ic}\
\overbrace{\int_{-1}^{1}\
\underbrace{\int_{-\infty}^{\infty}\expo{\ic\pars{\omega - \nu}t}
\,\,{\dd t \over 2\pi}}_{\ds{\delta\pars{\omega - \nu}}}\
\,\dd\omega}^{\ds{=\ 1}}\ \,\dd\nu
\\[5mm] &\
{\pi \over 2}\int_{-1}^{1}\expo{5\nu\ic}\,\dd\nu =
\pi\int_{0}^{1}\cos\pars{5\nu}\,\dd\nu =
\pi\,{\sin\pars{5} \over 5}
\\[5mm] = &\
\bbx{\pi\,\on{sinc}\pars{5}} \approx -0.6025 \\ &
\end{align}
