Some useful Fourier transform theorems

There are several basic theorems of Fourier transformation which we have to know in order to understand much of 

Eiichi Fukushiitta and Stephen B. W. Roeder, Experimental Pulse NMR A Nuts and Bolts Approach 

Copyright 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book Program. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. 

Before getting into the theorems, we have to remember what the transformation equations look like. They are reproduced here from Section I.D.2. 

The addition theorem states that if f(t) and g(t) have the Fourier transforms F(w) and G(w), then the function f(t)+g(t) has the Fourier transform F(uj)+G(tu). This follows easily from the definition of the transform. 

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The Fourier method is based on the central section theorem, which states that the Fourier transform of a projection is a central section in Fourier space. This means that projections at different angles then provide sections of Fourier space at these angles and thus the space can be filled up. We can thus obtain the complete three-dimensional Fourier transform of the object. The reverse Fourier transformation of such a volume will generate the three-dimensional density distribution of the object in real space. For particles with icosahedral or helical symmetry, a Fourier-Bessel transformation is widely used since the use of a cylindrical coordinate system may avoid some interpolation errors. 

As with the Fourier transform, we denote a Laplace transform by a capital letter and the function by a lowercase letter. The Laplace transform is similar to a one-sided Fourier transform, except that it has a real exponential instead of the complex exponential of the Fourier transform. If we consider complex values of the variables, the two transforms become different versions of the same transform, and their properties are related. The integral that is carried out to invert the Laplace transform must be carried out in the complex plane, and we do not discuss it. Fortunately, it is often possible to apply Laplace transforms without carrying out such an integral by applying some useful theorems. 2 We will discuss the use of Laplace transforms in solving differential equations in Chapter 12. 

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