Its complexity of n log 2n allows it to be used on large quantities of data, a significant improvement over the n2 complexity of its discrete counterpart.  Its applications include quickly multiplying large polynomials and compression analysis, but wavelets are starting to replace it in practical applications because of the Gibbs phenomenon at sharp transitions.  It is based on the discovery that it is possible to take any periodic function of time and resolve it into an
Its complexity of n log 2n allows it to be used on large quantities of data, a significant improvement over the n2 complexity of its discrete counterpart.  Its applications include quickly multiplying large polynomials and compression analysis, but wavelets are starting to replace it in practical applications because of the Gibbs phenomenon at sharp transitions.  It is based on the discovery that it is possible to take any periodic function of time and resolve it into an equivalent infinite summation of sine and cosine waves.  FTP, name this process of breaking a complex signal into an infinite series.
Its complexity of n log 2n allows it to be used on large quantities of data, a significant improvement over the n2 complexity of its discrete counterpart.  
Fast Fourier Transform (prompt on Fourier transform)