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One of the fundamental questions of long-standing theoretical interest is the complexity of fast Fourier transforms and the proof of lower bounds on the exact number of operations, and many open problems remain. Taking advantage of the existence of a generator for the modulo prime n multiplicative group, the Rader algorithm expresses a DFT of size n prime as a (composite) cyclic convolution of size n − 1; this can then be calculated with a pair of ordinary FFTs. Again, a strict lower bound has not been proven. convolution theorem (although Winograd uses other convolution methods). What is the lower bound on the complexity of fast Fourier transform algorithms? Inside the FFT Black Box: Series and Parallel Fast Fourier Transform Algorithms. Most attempts to reduce or prove the complexity of FFT algorithms have focused on the case of ordinary complex data because it is the simplest. Because this size does not fit into main memory, out-of-core FFTs are an active area of research.
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