Fast Fourier Transform

Time-space tradeoffs for computing functions, using connectivity properties of their circuits

Necessary and sufficient conditions for a direct sum of local rings to support a generalized discrete Fourier transform are derived. In particular, these conditions can be applied to any finite ring. The function O(N) defined by Agarwal and Burrus for transforms over ZN is extended to any finite ring R as O(R) and it is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) This result is applied to the homomorphic images of rings-of algebraic integers.

The Discrete Fourier Transform Over Finite Rings with Application to Fast Convolution

Two discrete Fourier transform based algorithms are proposed for the computation of the Moore-Penrose and Drazin inverse of a multivariable polynomial matrix.

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Inverses of multivariable polynomial matrices by discrete fourier transforms

The main purpose of this work is to provide an optimized version of the Newton divided-difference algorithm presented by [11] for the polynomial interpolation problem. This optimized algorithm is used for the computation of the determinant of a two-variable polynomial matrix.

On the Newton multivariate polynomial interpolation with applications

In the past decade the Cooley-Tukey fast Fourier transform (FFT) [1] has achieved the status of a “super” algorithm. As a numerical (complex field) algorithm, the FFT has revolutionized large scale time series analysis in a way that counts most—economic. (See, e.g., Refs. 3-6.) Since the late sixties, the FFT has also emerged as an important algebraic(abstract field) algorithm, with many interesting applications to the theory and practice of algebraic computing. The abstract character of the FFT, in particular its role as an algebraic algorithm, is what this paper is about.Our discussion centres around the following questions:1. What is the discreteFourier transform?2. What is the fastFourier transform?3. What is its role in algebraiccomputing?4. Is a finite field(mod p) FFT feasible?

John D. Lipson

Papers

The fast Fourier transform its role as an algebraic algorithm