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A fast Fourier transform is an efficient algorithm for working out the discrete Fourier transform - which itself is a Fourier transform on 'discrete' data, such as might be held on a computer. Contrast this to a 'continuous Fourier transform' on, say, a curve. One would need an infinite amount of data points to truly represent a curve, something that cannot be done with a computer.
Check out: The Scientist And Engineer's Guide To Digital Signal Processing. It is a free, downloadable book that deals, inter alia, with Fourier transforms; chapters 8-12are germane to your question. This is a highly practical, roll-yer-sleeves-up book for, as the title says, scientists and engineers, but Smith describes the underlying theory well. The sample code supplied with the book is in BASIC and FORTRAN, of all things; the author does this for didactic purposes to make the examples easy to understand rather than efficient.
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What is the difference between discrete fourier transform and fast fourier transform?
discrete fourier transformer uses digital signals whereas the fast fourier transform uses both analog and digital.
What is the difference between the fourier laplace transform?
They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform. The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.
Can you give an example of how Fast Fourier Transforms are used in financial models?
There is a beautiful paper by Ales Cerny entitled "Introduction to Fast Fourier Transform in finance", which gives many interesting examples.
Is there away to sort an array of data using the fast Fourier transform and finding the highest lower or average value finding his value or even best his position?
The Fast Fourier Transform is an implementation of the Discrete Fourier Transform. The DFT is a method of processing a time-sampled signal (eg, an audio wave) into a series of sines and cosines. As such, it is not a sorting algorithm, so this question does not make any sense.
What is the difference between Fourier transform and Laplace transform and z transform?
Fourier transform and Laplace transform are similar. Laplace transforms map a function to a new function on the complex plane, while Fourier maps a function to a new function on the real line. You can view Fourier as the Laplace transform on the circle, that is |z|=1. z transform is the discrete version of Laplace transform.
