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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/p m ), the unit group of Z/p m . Consider the space L(Z/p m ). For m > 1, we defined the space
$$L_{0}=L(1,m-1)$$
(1)
of M-decimated and M m−1 -periodic functions on Z/p m with M = p Z /p m and proved that
$$L(Z/p^{m})=W\bigoplus L_{0}$$
(2)
where W is the orthogonal complement of L0 in L(Z/p m ). The space L0 and W are invariant under the action of the Fourier transform F of Z/p m . 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/p m .

Keywords

Orthonormal Basis Discrete Fourier Transform Unit Group Orthogonal Basis Orthogonal Complement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Tolimieri, R. “Multiplicative Characters and the Discrete Fourier Transform”, Adv. in Appl. Math. 7 (1986), 344–380.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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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