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Can the divide and conquer nature of the FFT be generalized to other transforms (z Transform, chirp, etc) automatically?

Is there an algorithm that takes in a description of transform (I don't know what information would be needed) and can produce a fast FFT like function?

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While this is not a direct answer, the following discussion on MO (Mathoverflow) relates to finding convolution operations that admit fast FFT-like operations on other algebraic structures.

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