The Fourier Transform - what is it?
Well, suppose we have some sum or computation we wish to evaluate, for example to calculate a*b for some a and b, e.g. large integers (of several thousands of digits or more).
Sometimes it can be easier to map each of the inputs to a new domain via some transform, perform the multiplication (approximating suitably if necessary) and then apply the reverse transform to get the original answer. Such a transform is the "Fourier" transform (and its inverse), and because of the way it maps objects/events to wave functions (expressed in terms of sin's and cos's) (and back) its respective domains of operation are termed 'time' and 'frequency'.
http://en.wikipedia.org/wiki/Fourier_transform
http://mathworld.wolfram.com/FourierTransform.html
http://mathworld.wolfram.com/FourierSeries.html
Obviously, in order for the process to work, it depends on the reciprocity of the transforming function. This is achieved by the orthogonality of the underlying individual waveforms (Sturm-Liouville).
http://en.wikipedia.org/wiki/Sturm-Liouville_theory
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