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Multiplicative characters and the discrete Fourier transform
Multiplicative character theory will be used to reprove results from a paper of Auslander-Feig-Winograd (Adv. in Appl. Math.5 (1984), 31–55) on the multiplicative complexity of the discrete Fourier transform. The Fourier transform FA acting on the space L(A) of all complex-valued functions on <img height="18" border="0" style="vertical-align:bottom" width="88" alt="View the MathML source" title="View the MathML source" src="http://ars.els-cdn.com/content/image/1-s2.0-0196885886900394-si1.gif"> an integer, is decomposed relative to “rational” subspaces of L(A) naturally described by multiplicative characters of A. This decomposition is the main tool needed to understand the ad hoc constructions in the Auslander-Feig-Winograd (op cit.) paper.