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∈Cn by a matrix Pb,α∈Cn×n, whose (k,ℓ)th element is defined as pℓ(αk) for polynomials pℓ(x)∈C[x] from a list b={p0(x),…,pn-1(x)} and sample points αk∈C from a list α={α0,…,αn-1}. 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 O(n log n) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.