  A polynomial transform is the multiplication of an input vector $x\in\C^n$ by
a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is
defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list
$b=\{p_0(x),\dots,p_{n-1}(x)\}$ and sample points $\alpha_k\in\C$ from a list
$\alpha=\{\alpha_0,\dots,\alpha_{n-1}\}$. Such transforms find applications in
the areas of signal processing, data compression, and function interpolation.
Important examples include the discrete Fourier and cosine transforms. 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.
