Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction

A polynomial transform is the multiplication of an input vector x ∈ C by a matrix Pb;α ∈ Cn×n; whose ðk;lÞth element is defined as plðαkÞ for polynomials plðxÞ ∈ C1⁄2x from a list b 1⁄4 fp0ðxÞ; : : : ; pn−1ðxÞg and sample points αk ∈ C from a list α 1⁄4 fα0; : : : ;αn−1g. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. An important example includes the discrete Fourier transform. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel Oðn log nÞ general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.