Multiplicative Characters and the FFT

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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 (1) $$L_{0}=L(1,m-1)$$ of M-decimated and M m−1 -periodic functions on Z/p m with M = p Z /p m and proved that (2) $$L(Z/p^{m})=W\bigoplus L_{0}$$ 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 .