Continuous-Time Fourier Theorems
Selected Fourier theorems for the continuous-time case are stated and proved in Appendix B. However, two are sufficiently important that we state them here.
The scaling theorem (or similarity theorem) says 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: See §B.4.
The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page ). For this and other continuous-time Fourier theorems, see Appendix B.
Definition: A function is said to be of order if there exist and some positive constant such that for all .
Proof: See §B.18.
Fourier Theorems for the DTFT