# Rationality

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

## Abstract

Multiplicative character theory provides a natural setting for developing the complexity results of Auslander, Feig and Winograd [1]. The first reason for this is the simplicity of the formulas describing the action of FT on multiplicative characters. We will now discuss a second important property of multiplicative characters. In a sense defined below, the spaces spanned by certain subsets of multiplicative characters are rational subspaces. As a consequence, we will be able to rationally manipulate the FT matrix F(p m ) into block diagonal matrices where each block action corresponds to some polynomial multiplication modulo a rational polynomial of a special kind. This is the main result in the work of Auslander, Feig and Winograd. Details from the point of view of multiplicative character theory can be found in [2].

## Keywords

Rational Basis Discrete Fourier Transform Basis Matrix Vandermonde Matrix Multiplicative Character
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## References

1. [1]
Auslander, L., Feig, E. and Winograd, S. “The Multiplicative Complexity of the Discrete Fourier Transform”, Adv. in Appl. Math., 5, 1984, pp. 31–55.
2. [2]
Tolimieri, R. “Multiplicative Characters and the Discrete Fourier Transform”, Adv. in Appl. Math., 7, 1986, pp. 344–380.
3. [3]
Rader, C. “Discrete Fourier Transforms When the Number of Data Samples is Prime”, Proc. IEEE, 56, 1968, pp. 1107–1108.
4. [4]
Winograd, S. “Arithmetic Complexity of Computations”, CBMS Regional Conf. Ser. in Math. SIAM, 33, Philadelphia, 1980.Google Scholar
5. [5]
Tolimieri, R. “The Construction of Orthogonal Basis Diagonalizing the Discrete Fourier Transform”, Adv. in Appl. Math., 5, 1984, pp. 56–86.

## Authors and Affiliations

• Richard Tolimieri
• 1
• Chao Lu
• 2
• Myoung An
• 3
1. 1.Department of Electrical EngineeringCity College of CUNYNew YorkUSA
2. 2.Department of Computer and Information SciencesTowson State UniversityTowsonUSA
3. 3.A.J. Devaney AssociatesAllstonUSA