The Fourier transform General features

Fourier demonstrated that for any function/(x), there exists another function F(h) such that 

The Fourier transform operation is reversible. That is, the same mathematical operation that gives F(h) from f(x) can be carried out in the opposite direction, to givef(x) from F(h) specifically, 

Returning to the visual transforms of Figs. 2.7-2.10 each object (the sphere in Fig. 2.7, for instance) is the Fourier transform (the back-transform, if you wish) of its diffraction pattern. If we build a model that looks like the diffraction pattern on the right, and then obtain its diffraction pattern, we get an image of the object on the left. 

There is one added complication. The preceeding functionsf(x) and F(h)are one-dimensional. Fortunately, the Fourier transform applies to periodic functions in any number of dimensions. To restate Fourier s conclusion in three dimensions, for any function f(x,y,z) there exists the function F(h,k,l) such that 

Thinking again about the potential usefulness of computing IT s in crystallography, you will see that we can use the Fourier transform to obtain information about real space, f(x,y,z), from information about reciprocal space, F(h,k, 1)- Specifically, the diffraction pattern contains information whose Fourier transform is information about the contents of the unit cell. 

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