## Abstract

An algebraic theory for the discrete cosine transform (DCT) is developed, which is analogous to the well-known theory of the discrete Fourier transform (DFT). Whereas the latter diagonalizes a convolution algebra, which is a polynomial algebra modulo a product of various cyclotomic polynomials, the former diagonalizes a polynomial algebra modulo a product of various polynomials related to the Chebyshev types. When the dimension of the algebra is a power of 2, the DCT diagonalizes a polynomial algebra modulo a product of Chebyshev polynomials of the first type. In both DFT and DCT cases, the Chinese remainder theorem plays a key role in the design of fast algorithms.

Original language | American English |
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Pages (from-to) | 81-106 |

Number of pages | 26 |

Journal | Linear Algebra and Its Applications |

Volume | 266 |

Issue number | 1-3 |

DOIs | |

State | Published - 15 Nov 1997 |