Fourier transforms of the function

Note that by the definition of the Fourier spectrum presented here, which we have given as the Fourier transform of the function or data under consideration, we may consider the interferogram function to be the Fourier... 

In spectroscopy one frequently deals with functions of time, f(t), such as the position r(t) of an electric charge, or a time-varying dipole moment, fi(t), which lead to emission of electromagnetic radiation if the second time derivative is not vanishing. The frequency spectrum of the associated emission is obtained by Fourier transform of the function of time. If the absolute value of the function, /(t), is integrable over all times, —co < t < co, one defines the Fourier transform according to... 

We mention specifically the Fourier transform of the rth time derivative of a function, with r = 1, 2,. .., which may be expressed as the Fourier transform of the function, fit), times cor,... 

To determine the function x(t), it is necessary to find the solution of the integral of Eq. (3.6) for a known h(t) and measured y(t) functions. One proposed method of calculating x(t) assumes that the shape of the function x(t) is known beforehand and that the transfer function h(t) can be experimentally determined.158 A more general solution uses Fourier transforms. If Eq. (3.6) is rewritten with the Fourier transforms of the functions x(t), y(t), and h(t - t), which are denoted by Y, X, and H, then, using the convolution theorem ... 

The lineshape function which describes the absorption and dispersion modes of an unsaturated, steady-state NMR spectrum is proportional to the Fourier transform of the function MxID(t) (24, 25, 99)... 

Thus, a generic hermitian operator can be expressed in terms of the representation of the group Hn. The existence of this expansion is demonstrated in Appendix 1. There, we also prove that the functions B(g) have to be obtained as the inverse Fourier transform of the functions Bw (q,p) of the phase-space variables (q,p) = qi,. . ., qn,Pi, , Pn), which are associated to the quantum operators, B, via the Wigner transform [10]. Since we are considering hermitian operators of the form B X, IiD J, the coefficients in eq.(22) are... 

According to the convolution theorem (35), the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of the functions. Thus, if S(r) and G(2)(r) have been measured, the... 

However, from the point of view of linear response theory, the definitions (174) or (178) suffer from several drawbacks. Actually, the function X ( , tw) as defined by Eq. (174) is not the Fourier transform of the function X (, x), but a partial Fourier transform computed in the restricted time interval 0 < x < tw. As a consequence, it does not possess the same analyticity properties as the generalized susceptibility x( ) defined by Eq. (179). While the latter, extended to complex values of co, is analytic in the upper complex half-plane (Smoo > 0), the function Xi ( - tw) is analytic in the whole complex plane. As a very simple example, consider the exponentially decreasing response function... 

This shows that Fi a) is the Fourier transform of the function f x) e , if a.i is held constant. Applying Fourier inversion formula (2.6.4), we obtain... 

The functions written in capital letters in (4.2.13)-(4.2.16) are the Fourier transforms of the functions written in small letters in (4.2.5)-(4.2.8). The superscript s indicates that the nonlinear transfer functions K (o)i,. .., o> ) in (4.2.15) and (4.2.16) are the Fourier transforms of impulse-response functions with indistinguishable time arguments, where the causal time order ti > >r is not respected. These transfer functions are invariant against permutation of frequency arguments. Equivalent expressions for the Fourier transforms of impulse-response functions with time-ordered arguments cannot readily be derived. 

A linear filter performs a convolution of the input function with the Fourier transform of the filter transfer function. According to the convolution theorem (cf. Section 4.2.3) application of a filter in one domain corresponds to multiplication of the Fourier transform of the function to be filtered with the filter-transfer function. To filter a backprojection image, eqn (6.1.3) is Fourier transformed,... 

In the treatment of EXAFS data, it is useful to obtain an inverse Fourier transform of the function (R) over a limited range of R. This procedure determines the contribution to EXAFS arising from shells of atoms within that range of R. If we consider a range of R from R - AR to R + AR, the inverse transform is given by the integral... 

Plots of the function K xUO vs. K at 100°K for the extended fine structure beyond the osmium edge for pure metallic osmium, and for the osmium-copper clusters in the catalyst containing 2 wt% Os and 0.66 wt% Cu, are shown in the left-hand sections of Figure 4.13. The associated Fourier transforms of the functions are shown in the right-hand sections of the figure. As previously noted, the Fourier transform yields the function n(R), the peaks of which are displaced from the true interatomic distances because of the phase shifts. Similar plots for the extended fine structure beyond the copper K edge for pure metallic copper and for the osmium-copper catalyst are given in our 1981 paper (32). 

Figure 1. Fast Fourier transform of the function g(t) =

The scattering function Ss(q, co) is simply the space-time Fourier transform of the function Gs(t, t) introduced in Section 5.4 or equivalently the time-Fourier transform of Fs (q, t) defined in the same section. The expressions for light scattering corresponding to Eq. (15.2.1) are derived in Section 5.4. Note that the light scattering is incoherent in this case because of the assumed lack of correlation between the space-time positions of the scatterers. 

Using (B.18), which states that the Fourier transform of the derivative of a function is iq times the Fourier transform of the function, (7.46) can be written also as... 

Show that the inverse Fourier transform of the function Z(s) defined by (C. 10) and (C.l 1) indeed leads to the lattice function z(r) given by (C.5), ignoring the exact value of the proportionality constant. 

A function that is compact in momentum space is equivalent to the band-limited Fourier transform of the function. Confinement of such a function to a finite volume in phase space is equivalent to a band-limited function with finite support. (The support of a function is the set for which the function is nonzero.) The accuracy of a representation of this function is assured by the Whittaker-Kotel nikov-Shannon sampling theorem (29-31). It states that a band-limited function with finite support is fully specified, if the functional values are given by a discrete, sufficiently dense set of equally spaced sampling points. The number of points is determined by Eq. (26). This implies that a value of the function at an intermediate point can be interpolated with any desired accuracy. This theorem also implies a faithful representation of the nth derivative of the function inside the interval of support. In other words, a finite set of well-chosen points yields arbitrary accuracy. 

In order to find the molecule-molecule scattering length as a function of the mass ratio M/m it is again more convenient to transform the integral equation for the function fir, R) into an equation in the momentum space. Introducing the Fourier transform of the function/(r,R)as/(k,p) = / d rd i /(r,R)exp(/k r/a+ /Pp R/fl), we obtain the following momentum-space equation ... 

Equation 1.12 is a Fourier transform of the function g to the function p, where is a function of the Cartesian coordinates x, y, and z, and time t. The function g may be obtained in a corresponding Fourier transformation of T T consists of an unimportant complex exponential and a contour function that defines the wave packet, moving with the velocity and acceleration of the particle. 

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