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The discrete Fourier transform: A canonical basis of eigenfunctions (0808.3214v1)
Published 23 Aug 2008 in cs.IT, cs.DM, math.IT, and math.RT
Abstract: The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the "discrete oscillator transform" (DOT for short). Finally, we describe a fast algorithm for computing the DOT in certain cases.