Inversion theorem

HanP81 Hanlon, P. A cycle-index sum inversion theorem. J. Comb. Theory A30 (1981) 248-269. 

The Fourier transform equations show that the electron density is the Fourier transform of the structure factor and the structure factor is the Fourier transform of the electron density. Examples are worked out in Figures 6.14 and 6.15. If the electron density can be expressed as the sum of cosine waves, then its Fourier transform corresponds to the sum of the Fourier transforms of the individual cosine waves (Figure 6.16). The inversion theorem states that the Fourier transform of the Fourier transform of an object is the original object, hence the opposite signs in Equations 6.12.1 and 6.12.2. This theorem provides the possibility of using a mathematical expression to go back and forth between reciprocal space (structure factors) and real space (electron density), so that the phrase and vice versa is applicable here. 

Once the Laplace transform u(x,s) of the temperature () (x, /,) which fits the initial and boundary conditions has been found, the back-transformation or so-called inverse transformation must be carried out. The easiest method for this is to use a table of correspondences, for example Table 2.3, from which the desired temperature distribution can be simply read off. However frequently u(x,s) is not present in such a table. In these cases the Laplace transformation theory gives an inversion theorem which can be used to find the required solution. The temperature distribution appears as a complex integral which can be evaluated using Cauchy s theorem. The required temperature distribution is yielded as an infinite series of exponential functions fading with time. We will not deal with the application of the inversion theorem, and so limit ourselves to cases where the inverse transformation is possible using the correspondence tables. Applications of... 

Inverse transformation to the time region (u — , s — t) using a correspondence table (u —o ) or the general inversion theorem. 

The inverse transformation can be achieved using the inversion theorem, which we do not need here, as we are only interested in the temperature at the point x = 0. For this we find... 

The application of the Laplace transformation delivers the same result. The inverse transformation from the frequency to the time region requires the use of the inversion theorem, see 2.3.2. In order to avoid this in this case the simple, classical product solution is applied. 

Solutions of (2.223) for other power densities W(x,t) and boundary conditions can be found in the same way. However, for bodies of finite thickness, the inverse transformation of u(x, s) can normally only be carried out using the inversion theorem mentioned in section 2.3.2. For cases such as these see [2.1] and the detailed examples discussed in [2.26]. 

All definitions are as before and F is a constant. The solution obtained by use of the Laplace transform and Inversion theorem is... 

Now, let us look at the inverse theorem. We have two operators and any eigenvector of A is also an eigenvector of B. We want to show that the two operators commute. Let us write the two eigenvalue equations Aij/n = and Btlr = bn n- Let us take a vector 0. Since the eigenvectors form the complete set, then... 

It is important to stress that this curve encloses no singularities. In fact, it is the act of enclosing a singularity (or many singularities) that leads to a very simple, classical result for solving the Laplace Inversion theorem. 

The exploitation of Cauchy s First theorem requires us to test the theorem for exceptional behavior. This allows, as we shall see, direct applications to the Laplace Inversion theorem. 

The final elementary component of complex analysis necessary to effect closure of the Inversion theorem (Eq. 9.3) is Residue theory. 

At the beginning of this chapter, we quoted the Mellin-Fourier Inversion theorem for Laplace transforms, worth repeating here... 

It may now be clear why the factor 1 /Ivi appears in the denominator of the Inversion theorem. It should also be clear that... 

We shall first consider the Inversion theorem for pole singularities only. The complex function of interest will be fis) = e F(s). The contour curve, denoting selected values of s, is called the First Bromwich path and is shown in Fig. 9.4. 

The final simplified result for the Inversion theorem when pole singularities exist in F(s) is recapitulated as... 

Use the Inversion theorem to find fit) corresponding to the Laplace transform... 

We have derived the general Inversion theorem for pole singularities using Cauchy s Residue theory. This provides the fundamental basis (with a few exceptions, such as /s) for inverting Laplace transforms. However, the useful building blocks, along with a few practical observations, allow many functions to be inverted without undertaking the formality of the Residue theory. We shall discuss these practical, intuitive methods in the sections to follow. Two widely used practical approaches are (1) partial fractions, and (2) convolution. 

Dividing this by 2vi yields the Fourier-Mellin Inversion theorem, which defines the inverse transform of /7... 

APPENDIX C Derivation of the Fourier-Mellin Inversion Theorem written for the problem at hand as... 

These are the two building blocks to prove the Fourier-Mellin inversion theorem for Laplace transforms. 

However, all three techniques share several common features. First, they measure local, not average, surface properties. Any theory must therefore include the local surface properties if it is to be useful. Second, they all lack a simple inversion theorem in no case is it possible to infer directly physical properties of the system from the scanning probe results. Interpretation therefore has to proceed by an indirect interpretation cycle ... 

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