Because the inverse transform of a transform returns the original function, this allows a definition of an interesting function called the Dirac delta function . Combining the two equations in Eq. 185 into a single equation, and then interchanging the order of integration:
(188) 
(189) 
(1810) 
The delta function can be used to derive an important conservation theorem.
If represents the density of some function (i.e., a wavefunction like ), the squaremagnitude of integrated over all of space should be the total amount of material in space.
(1811) 
(1812) 
(1813) 
The proof is straightforward that the convolution of two functions, and , is a Fourier integral over the product of their Fourier transforms, and :
(1814) 
Another way to think of this is that ``the net effect on the spatial function due two interfering waves is contained by product the fourier transforms.'' Practically, if the effect of an aperture (i.e., a sample of only a finite part of real space) on a wavefunction is desired, then it can be obtained by multiplying the Fourier transform of the aperture and the Fourier transform of the entire wavefunction.
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Creating Lattices for Subsequent Fourier Transform
A diffraction pattern from a group of scattering centers such atoms is related to the Fourier transform of the ``atom'' positions:

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Discrete Fourier Transforms
A Fourier transform is over an infinite domain. Numerical data is seldom infinite, therefore a strategy must be applied to get a Fourier transform of data. Discrete Fourier transforms (DFT) operate by creating a lattice of copies of the original data and then returning the Fourier transform of the result. Symmetry elements within the data appear in the Discrete Fourier transform and are superimposed with the Transform of the symmetry operations due to the virtual infinite lattice of data patterns. Because there are a finite number of pixels in the data, there are also the same finite number of subperiodic wavenumbers that can be determined. In other words, the Discrete Fourier Transform of a image will be a data set of wavenumbers:
representing the amplitudes of the indicated periodicities.

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Fourier Transforms on Lattices with Thermal Noise
Lattices in real systems not only contain defects, but also some uncertainty in the positions of the atoms because of thermal effects such as phonons.

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Imaging from Selected Regions of Reciprocal Space
To select and interpret different regions of Fourier space, a function will be produced that selects a particular region of the Fourier Space (i.e., as selected set of possible periodicities) and then visualize the BackTransform of only that region.

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Taking Discrete Fourier Transforms of Images
A image in graphics format, such as a .gif, contains intensity as a function of position. If the function is grayscale data, then each pixel typically takes on discrete gray values between 0 and 255. This data can be input into MATHEMATICA and then Fourier transformed.
