I've been trying to find Fourier transform of that function.
What I got so far: - The function is even, so the FT is of order $2$, and an involution. - When we integrate the FT of that function over $\mathbb{R}$, we get $1$ (because FT of FT is the function itself, we put argument$=0~$ and $$\lim_{x\to 0} \frac{\sin (x)}{x}=1$$ well known.
But I don't know how to precisely calculate FT of that function given as a combination/mix of elementary functions. I tried also changing variables etc. yesterday but nothing worked. Thanks in advance for help.
Let's first define $\operatorname{sgn}t$ as $1$ if $t>0$, $-1$ if $t<0$ and $0$ if $t=0$ so (by this and this)$$\int_{\Bbb R}\frac{\sin tx}{x}dx=\pi\operatorname{sgn}t.$$
Because odd functions integrate to 0, $$\int_{\Bbb R}\frac{\sin x}{x}\exp ikxdx=\int_{\Bbb R}\frac{\sin x}{x}\cos kxdx=\frac12\int_{\Bbb R}\frac{\sin[(1+k)x]+\sin[(1-k)x]}{x}dx\\=\frac{\pi}{2}\left(\operatorname{sgn}(1+k)+\operatorname{sgn}(1-k)\right)=\pi1_{(-1,\,1)}(x).$$