Fourier transform shift theorem

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

Now in view of the shift theorem for Fourier transforms as applied to one-sided Fourier transforms [38], namely. 

Hence using the shift theorem for Fourier transforms... 

The Fourier transform of (7.50) is obtained with the help of the shift theorem (see Appendix B), which shows that the Fourier transform of 8(—z — t) is equal to exp(z ) times the Fourier transform of < (—z). We thus obtain... 

The shift theorem states that if the function f(t) has the Fourier transform F(w), then the function f(t-a) has the transform F(uj) exp(-iu)a). Its derivation is also quite simple. We go through it here for illustration. 

This theorem means that if a time domain signal is shifted in time, its Fourier transform remains identical except that it... 

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