Chapter

Algorithms for Discrete Fourier Transform and Convolution

Part of the series Signal Processing and Digital Filtering pp 295-321

Multiplicative Characters and the FFT

  • R. TolimieriAffiliated withCenter for Large Scale Computing, City University of New York
  • , Myoung AnAffiliated withCenter for Large Scale Computing, City University of New York
  • , Chao LuAffiliated withCenter for Large Scale Computing, City University of New York

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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 .