Gaussian distribution Fourier transform

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

FIGURE 7.11 One-component injection detected by SCOFT. (a) Time-domain data produced when a fluorescein sample is injected down the separation channel 55 peaks, spaced at 0.58 s, superimposed on the Gaussian distribution of the expanded laser beam, (b) Fourier transformation of part (a). Peaks for 1.74 Hz (fundamental) and 3.43 Hz (second harmonic) can be seen. The frequency of 1.74 Hz was correlated to the migration time of 0.58 s [698]. Reprinted with permission from the American Chemical Society. 

In conclusion, the ordinary central limit theorem (CLT) establishes that all the propagators sharing the same second moment (c2) yields in the time asymptotic limit the same gaussian pdf. What is the property that the propagators with a diverging second moment must have in common to produce the same pdf in the time asymptotic limit The answer to this question is equivalent to establishing the GCLT [45]. The answer to this important question rests on the anti-Fourier transform of the expression of Eq. (102), which turns out [46] to yield for v oc a distribution proportional to 1/ E,, with... 

This is the Fourier transform of an n-dimensional Gaussian distribution with the mean values... 

For a point charge, the Fourier transform of the charge density is p(k) = Ze, the charge itself. The Fourier transform of a Gaussian charge distribution is... 

The energy of the photon remains unchanged, i.e. the diffraction is elastic. The Fourier transform of a crystal is obtained by introducing equation (3.39) into (3.35). This operation requires a deeper study of the behavior of when N is large, a function which is composed of a set of sharp peaks (Fig. 3.5). The distribution is a mathematical representation of an infinitely sharp peak. S may be considered to be the limit of a Gaussian function... 

Calculate the thermodynamic average of the surface-averaged mean curvature, H, the mean-square curvature, and the Gaussian curvature, K, of the surface if the Fourier transform of the random" field, rjr(q) is described by a Gaussian probability distribution, P [ (5) ]- Perform the calculation for the case where the different wavevectors, q, are uncoupled so that one can write the probability distribution for a single mode as... 

Gaussian filters reflect a Gaussian distribution with mean x and variance They are the most widely used filters in present-day LES research, because they provide a good balance for transforming between the physical space and the Fourier... 

When monochromatic light is scattered by moving particles that show thermal motion, the field amplitudes E oo) show a Gaussian distribution. The experimental arrangement for measuring the homodyne spectrum is shown in Fig. 7.31. The power spectrum P (jo) of the photocurrent (7.68), which is related to the spectral distribution I oo), is measured either directly by an electronic spectrum analyzer, or with a correlator, which determines the Fourier transform of the autocorrelation function C(r) a i t)) i t -f- r)). According to (7.63), C(r) is related to the intensity correlation function G (r), which yields (7.64), and I co). 

In the previous paragraphs it was pointed out that a discrete time random walk, or a CTRW with a finite hrst moment for the waiting time distribution, on a lattice with a fixed jump distance led to a Gaussian diffusion process with a probability density given by Eq. (84). The spatial Fourier transform of this equation is... 

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