I have read on wikipedia that:
The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency.
Since the argument is the phase offset, I thought this:
If f(x) and g(x) = f(ax) Basically, the g function is the x function squeezed by a factor a. I would think that the frequencies that make it up, are the same as the one of the f function but multiplied by a. Therefore: $$ \hat{h}(\xi)= \hat{f}(\frac{\xi}{a}) $$ But wikipedia says:
Time Scaling For a non-zero real number a, if h(x) = f(ax), then $$ \hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right). $$
Can somebody explain why the amplitude seems also changed , as we divide by a ?