Multiplicative Characters and the FFT

  • R. Tolimieri
  • Myoung An
  • Chao Lu
Part of the Signal Processing and Digital Filtering book series (SIGNAL PROCESS)

Abstract

Fix an odd prime p throughout this chapter, and set U(m) ≡ U(Z/pm), the unit group of Z/pm. Consider the space L(Z/pm). For m > 1, we defined the space
$$L_{0}=L(1,m-1)$$
(1)
of M-decimated and Mm−1 -periodic functions on Z/pm with M = pZ/pm and proved that
$$L(Z/p^{m})=W\bigoplus L_{0}$$
(2)
where W is the orthogonal complement of L0 in L(Z/pm). The space L0 and W are invariant under the action of the Fourier transform F of Z/pm. The action of F on L0 was described in the preceeding chapter. We will now take up the action of F on W. For this purpose, we introduce the multiplicative characters on the ring Z/pm.

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References

  1. [1]
    Tolimieri, R. “Multiplicative Characters and the Discrete Fourier Transform”, Adv. in Appl. Math. 7 (1986), 344–380.MathSciNetMATHCrossRefGoogle Scholar
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    Auslander, L. Feig, E. and Winograd, S. “The Multiplicative Complexity of the Discrete Fourier Transform”, Adv. in Appl. Math. 5 (1984), 31–55.MathSciNetMATHCrossRefGoogle Scholar
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    Rader, C. “Discrete Fourier Transforms When the Number of Data Samples is Prime”, Proc. of IEEE, 56 (1968), 1107–1108.CrossRefGoogle Scholar
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    Winograd, S. Arithmetic Complexity of Computations, CBMS Regional Conf. Ser. in Math. Vol. 33, Soc. Indus. Appl. Math., Philadelphia, 1980.Google Scholar
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    Tolimieri, R. “The Construction of Orthogonal Basis Diagonalizing the Discrete Fourier Transform”, Adv. in Appl. Math. 5 (1984), 56–86.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • R. Tolimieri
    • 1
  • Myoung An
    • 1
  • Chao Lu
    • 1
  1. 1.Center for Large Scale ComputingCity University of New YorkNew YorkUSA

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