Fourier convolution theorem

The last relation in equation (Al.6.107) follows from the Fourier convolution theorem and tlie property of the Fourier transfonn of a derivative we have also assumed that E(a) = (-w). The absorption spectmm is defined as the total energy absorbed at frequency to, nonnalized by the energy of the incident field at that frequency. Identifying the integrand on the right-hand side of equation (Al.6.107) with the total energy absorbed at frequency oi, we have... 

In order to deduce Scherrer s equation first an infinite crystal is considered that is, second, restricted (i.e multiplied) by a shape function (cf. p. 17). Thus from the Fourier convolution theorem (Sect. 2.7.8) it follows that in reciprocal space each reflection is convolved by the Fourier transform of the square of the shape function - and Scherrer s equation is readily established. 

This fact has been discussed in Sect. 2.7.5, p. 24 on the basis of the Fourier convolution theorem (Sect. 2.7.8). 

Making use of equation (15) and (22), together with the Fourier convolution theorem, we can see that... 

To obtain the corresponding structure factor expression, the Fourier convolution theorem is applied, or... 

According to the Fourier convolution theorem, further discussed in section 5.1.3, the Fourier transform of the convolution in expression (2.14) is the product of the Fourier transforms of the individual functions, or... 

It is of importance that expression (5.12) holds even when /(x) is known only in part of space, as is the case in a crystallography experiment at finite resolution determined by Hmax. Using the Fourier convolution theorem, we may write (Dunitz and Seiler 1973)... 

The derivation of the electrostatic properties from the multipole coefficients given below follows the method of Su and Coppens (1992). It employs the Fourier convolution theorem used by Epstein and Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A direct-space method based on the Laplace expansion of 1/ RP — r has been described by Bentley (1981). 

In Eq. (8.18), we wrote the potential as a convolution of the total density and the operator 1 /r. Similarly, the integrals encountered in the evaluation of the peripheral electronic contributions to Eqs. (8.35) (8.37) are convolutions of the electron density p(r) and the pertinent operator. They can be evaluated with the Fourier convolution theorem (Prosser and Blanchard 1962), which implies that the convolution of /(r) and p(r) is the inverse transform of the product of their... 

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

Thanks to the space and time convoluted nature of Eq. (170) and to the Laplace-Fourier convolution theorem, we get the counterpart of Eq. (62), which reads... 

By virtue of the Fourier convolution theorem, this leads to a so-called Debye-Waller factor, DW(Q) that modulates the scattering at high Q ... 

---