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An Intuitive Explanation of Fourier Theory
Fourier theory is pretty complicated mathematically. But there are some beautifully simple holistic concepts behind Fourier theory which are relatively easy to explain intuitively. There are other sites on the web that can give you the mathematical formulation of the Fourier transform. I will present only the basic intuitive insights here, as applied to spatial imagery.
preview: http://cns-alumni.bu.edu  

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Fast Fourier Transform (FFT)
In this section we present several methods for computing the DFT efficiently. In view of the importance of the DFT in various digital signal processing applications, such as linear filtering, correlation analysis, and spectrum analysis, its efficient computation is a topic that has received considerable attention by many mathematicians, engineers, and applied scientists.
preview: http://www.cmlab.csie.ntu.edu.tw  

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Fast Fourier Transform @ Mathworld
The fast Fourier transform (FFT) is a discrete Fourier transform algorithm which reduces the number of computations needed for points from to , where lg is the base-2 logarithm. If the function to be transformed is not harmonically related to the sampling frequency, the response of an FFT looks like a sinc function (although the integrated power is still correct). Aliasing (leakage) can be reduced by apodization using a tapering function....
preview: http://mathworld.wolfram.com  

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Fast Fourier transform @ Wikipedia
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked...
preview: http://en.wikipedia.org  

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Fourier Transform @ Wikipedia
In mathematics, the Fourier transform, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. A useful analogy is the relationship between a series of pure notes (the...
preview: http://en.wikipedia.org  

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Introduction to Fourier Transforms For Image Processing
The Fourier Transform ( in this case, the 2D Fourier Transform ) is the series expansion of an image function ( over the 2D space domain ) in terms of 'cosine' image (orthonormal) basis functions.
preview: http://www.cs.unm.edu  

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The Fourier Transform
As you know, sound is vibration that propagates through air (or any other medium; don't be fooled by cheap science fiction movies--sound cannot cross the vacuum of empty space). What causes the vibration of the air is usually the vibration of other objects--vocal chords, musical instruments, speakers, and so on.
preview: http://www.relisoft.com