Fourier transform function, definition

The definition (21) of the TS trajectory in white noise can be rewritten as follows [40]. For a function /(f) with the Fourier transform... 

The Fourier transform introduces the wavenumber vector , which has units of 1 /length. Note that, from its definition, the velocity spatial correlation function is related to the Reynolds stresses by... 

Easily proved from the definition of the Fourier transform, this theorem states that convolving two functions is equivalent to finding the product of their Fourier transforms. Specifically, if a(x), h(x), and g(x) have transforms A(oo), B(co), and G(co), then... 

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... 

We will mainlv be concerned with the Fourier transform of the correlation functions. We use the definition... 

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... 

The characteristic function of this diffusion process is by definition the Fourier transform of p(x, t), namely... 

The two-point one time correlation functions, in the form presented in the preceding discussion, are not suitable for analyzing motions at different scales and specifically they are not suitable for understanding relations between movements of fluid characterized by different length and time scales. That is why it is better to use the 3D Fourier transforms of two-point correlations and to decompose them into waves of different frequencies or wave numbers. Turbulence has by definition a 3D character so it is obvious that the spectrum has to be 3D as well, to characterize turbulence properly. The ID spectrum of Taylor (see, e.g., [66]) oversimplifies the observed features of turbulence and may give misleading interpretations of the 3D held (see also, [113], p. 18). The differences and consequences of ID and 3D spectrum analysis are discussed by Hinze ([66], sects. 1-12 and 3-4) and Pope ([121], sect. 6.5). 

The study of the intensity distribution can be achieved by two methods either by fitting this peak with a calculated function or by a direct analysis using the Fourier transform of that peak. The fitting requires prior knowledge of the profile, which, by definition, can never exactly correspond to the experimental profile, and the Fourier analysis can only be achieved if the peaks do not overlap. These aspects will be discussed further in Part 2 of this book, which deals with microstrucmral analysis. 

There is a definite possibility of using classical physical quantities under operator companions, that is, an approach to quantization [ 19]. If a classical quantity was expressed by a function// , q) of the canonical variables, p, q, the Fourier transform of/can be used. Then,/is back-transformed from by... 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

The appearance of Avogadro s number in this expression results from the definition (7.2.57) of the constant K. This can be justified as follows. In ordinary space, the correlation function g(r) — 1 differs practically from zero only in a range of the same order as the size R of the molecules, let us say for instance 10 nm. Therefore, the Fourier transform //n(0) is a volume of the order of 103 nm3, but the product //n(0) is a macroscopic volume ( 104 cm3) which can be more directly measured, for instance, by looking at the osmotic pressure. 

The addition theorem states that if f(t) and g(t) have the Fourier transforms F(w) and G(w), then the function f(t)+g(t) has the Fourier transform F(uj)+G(tu). This follows easily from the definition of the transform. 

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