Algebraic Signal Processing Theory: Cooley–Tukey-Type Algorithms for Polynomial Transforms Based on Induction
Aliaksei Sandryhaila, Jelena Kovačević, Markus Püschel · SIAM Journal on Matrix Analysis and Applications · 2011
A polynomial transform is the multiplication of an input vector [Formula: see text] by a matrix [Formula: see text] whose [Formula: see text]th element is defined as [Formula: see text] for polynomials [Formula: see text] from a list [Formula: see text] and sample points [Formula: see text] from a list [Formula: see text]. 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 [Formula: see text] general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.