A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval $[ - \pi ,\pi ]$. The schemes of this paper are based on a combination of certain analytical considerations with the classical fast Fourier transform and generalize both the forward and backward FFTs. Each of the algorithms requires $O(N\cdot \log N + N\cdot \log (1/\varepsilon ))$ arithmetic operations, where $\varepsilon $ is the precision of computations and N is the number of nodes. The efficiency of the approach is illustrated by several numerical examples.
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