Spatial Fourier transform

In order to proceed with the method outlined above let us note, by taking the spatial Fourier transform of Eq. (8), that... 

We analyze the fringes by taking the spatial Fourier transform of the absorption image and measuring the area of the sidelobes as a function of the excitation frequency. We observe clear resonances at the predicted excitation energy, with an unprecedented signal to noise (Fig. 12). 

Spatial Fourier transformation of the self-correlation function (Eq. (26.7)) yields the intermediate scattering function. 

Anomalous diffusion of a continuous concentration field can be modelled in terms of fractional differential equations. To see how they arise we can write Eq. (2.9) for normal diffusion in terms of the spatial Fourier transform of the concentration field C(k, t). This can be easily done under periodic boundary conditions or in unbounded space as... 

The density fluctuation spectrum can be obtained by taking a spatial Fourier transform and a temporal Laplace transform of the three coupled equation (A3.3.18), equation (A3.3.19) and equation (A3.3.20), and then solving the resulting linear coupled algebraic set for the density fluctuation spectrum. (See details in the books by Beme-Pecora and Boon-Yip.) The result for p(, x ) given below is proportional to its frequency integral... 

A linear stability analysis of (A3.3. 57) can provide some insight into the stmcture of solutions to model B. The linear approximation to (A3.3.57) can be easily solved by taking a spatial Fourier transform. The result for the Ml Fourier mode is... 

Equation (3.2.3) can be expressed in terms of the spatial Fourier transform of the dielectric fluctuation... 

It is therefore often assumed that Gs is Gaussian. The spatial Fourier transform of this function is the intermediate scattering function [cf. Eq. (5.4.5)]... 

The spatial Fourier transform of this function is easily obtained and is... 

Substitution of Eq. (9.2.13) into Eq. (9.2.12), followed by a spatial Fourier transformation of the resulting equation, gives the two coupled diffusion eqd tions... 

An interesting feature of these conservation equations emerges from a consideration of the spatial Fourier transform of Eq. (10.3.10), which is... 

The double differential scattering cross section, according to (8.15), is proportional to the space-time Fourier transform of G(r,f). This is analogous to the fact, discussed in Section 1.5.2, that in the static approximation the intensity I(q) (or the differential scattering cross section ds/dQ) is given by the spatial Fourier transform of the density-density autocorrelation function defined in (1.79). In the special case of t = 0, G(r,0) denotes the probability of finding a particle at r when there is already a particle at position 0. G(r,0) is therefore related to the pair distribution function g(r) discussed in Section 4.1.1, as in... 

---