Is there any way to visualize a Fourier transform? I mean, if I am given a function, can I roughly guess what its Fourier transform should look line before doing the mathematics?
Suppose I am given the standard function, $f(x)=1$ for $|x|<1$. Its Fourier transform is of the form $\dfrac{\sin(k)}{k}$.
I want to know: can I guess this nature of the transform roughly using any logical arguments before doing the mathematics? I mean,the positions of zeros of the transform,or it will decay or not at higher value of k,and things like that.