Relationship between Paley-Wiener theorem and the stationary phase method?
Manell E. Zakharia · 2005
For a given complex functionz(\omega) = a(\omega) \exp (i \psi(\omega)), the Paley-Wiener theorem provides necessary and sufficient conditions, on the magnitudea(\omega), that there exists an "appropriate phase"\psi(\omega)such thatZ(t), the Fourier transform ofz(\omega), is a causal function. The theorem does not give a general way to derive this phase; it only proves its existence. The Fourier transform can also be calculated using asymptotic expansions and approximations such as the stationary phase method. That method can lead to a better knowledge of the conditions on the phase\psi(\omega)to be an "appropriate" one.