The scaling theorem (or similarity theorem) provides that if you horizontally ``stretch'' a signal by the factor in the time domain, you ``squeeze'' and amplify its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship.
Theorem: For all continuous-time functions possessing a Fourier transform,
and is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:
Proof: Taking the Fourier transform of the stretched signal gives
The absolute value appears above because, when , , which brings out a minus sign in front of the integral from to .
Differentiation Theorem Dual