### Abstract

A polynomial transform is the multiplication of an input vector x ∈ ℂ^{n} by a matrix P_{b α} ∈ C ^{n×n}, whose (k; l)th element is defined as p_{l}(α _{k}) for polynomials pl(x) ∈ ℂ [x] from a list b = {p _{0}(x);..., p_{n-1}(x)} and sample points α _{k}∈ℂ from a list α - {a_{0}...., α_{n-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.

Original language | English (US) |
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Pages (from-to) | 364-384 |

Number of pages | 21 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - 2011 |

### Keywords

- Algebra
- Discrete Fourier transform
- Discrete cosine transform
- Discrete sine transform
- Fast Fourier transform
- Fast algorithm
- Matrix factorization
- Module
- Polynomial transform

### ASJC Scopus subject areas

- Analysis

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## Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*32*(2), 364-384. https://doi.org/10.1137/100805777