Convolution relations approach

The multiplication of the spectrum by a window is equivalent to a convolution in the time domain, and hence the approach is related to the Savitzky-Golay procedure. Indeed, by (4.10) this latter is also a convolution of the function values and the coefficients c /F ... 

An alternative approach we shall use to analyze the further equation of a set results in more simple relations. Its idea is to rewrite the convolution (<71,1 X ) in equation (7.1.30) as an average over normalized distributions II , 1 of infinite series of products of the correlation functions and /. (As earlier, a mean value of the product is replaced by a product of mean values.) Finally, one gets... 

It is now well understood that this is not an approximation rather it is a way to force an equation with infinite memory to become compatible with Levy diffusion. The assumption (152) makes it possible for us to get rid of the time convolution nature of the generalized diffusion equation (133). At the same time, this key relation replaces the correlation function (t) with its second-order derivative and, as a consequence of Eq. (147), with the waiting time distribution /( ). This fact is very important. In fact, any Liouville-like approach makes the correlation function 3F(f) enter into play. The CTRW is a perspective resting on trajectories and consequently on /(f). Establishing a connection between the two pictures implies the conversion of 4> (f) into j/(r), or vice versa. Here, this conversion has been realized paying the price of altering the physics of the generalized diffusion equation (133). 

The effect of absorption in the specimen. The absorption coefficient corresponds to that of graphite. Sizes of the receiving slit length 0.1 mm, width 0.25 mm. The calculation according to the proposed method and the convolution approach are shown as a solid line and open circles, respectively. The vertical line at 2(p — 20° represents the Bragg angle to which the aberration function is related. (Reprinted from Ref. 53. Permission of the International Union of Crystallography.)... 

In the nonparametric approach, the input-output relation is represented either analytically (in convolutional form through Volterra-Wiener expansions where the unknown quantities are kernel functions). 

A more general approach to materials exhibiting frequency-dependent heat capacities is that of Schawe and co-workers [18]. For some linear (or at least, for temperature not varying too much, approximated linear) process, the rate of heat intake and rate of change of temperature can be related through a convolution. 

It is now desirable to deal with the nonclassical behavior of the kernel in the linear laws in a precise, formal way. Of course, one could simply try to improve the crude method just discussed such an approach is perfectly valid. However, we feel that an alternate procedure, which has almost always been used in the literature, is preferable. Mori s method allows the writing of equations with well-behaved kernels if the proper set of variables is chosen. The kernel in the linear laws is badly behaved due to the influence of the nonlinear variable. If we include the linear and nonlinear variables in the set of variables to which Mori s method is applied, the random forces and the dissipative fluxes (/ will be defined precisely in this section) will be projected orthogonal to all of these variables. The kernels in the resulting equations, the nonlinear Langevin equations, should behave classically. Thus, convolutions involving K will be converted into scalar multiplication by the classical relation. 

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