# Rationality

• R. Tolimieri
• Myoung An
• Chao Lu
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- 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 the subsets V k , 1 ≤ k > m, 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-Winograd. Details from the point of view of multiplicative character theory can be found in [2].

## Keywords

Rational Basis Discrete Fourier Transform Unit Group Basis Versus Rational Vector
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.

## References

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

## Authors and Affiliations

• R. Tolimieri
• 1
• Myoung An
• 1
• Chao Lu
• 1
1. 1.Center for Large Scale ComputingCity University of New YorkNew YorkUSA