ON ALGEBRAS RELATED TO THE DISCRETE COSINE TRANSFORM

An algebraic theory for the discrete cosine transform (DCT) is develop ed, which is analogous to the well-known theory of the discrete Fourie r transform (DFT). Whereas the latter diagonalizes a convolution algeb ra, which is a polynomial algebra module a product of various cyclotom ic polynomials, the former diagonalizes a polynomial algebra module a product of various polynomials related to the Chebyshev types. When th e dimension of the algebra is a power of 2, the DCT diagonalizes a pol ynomial algebra module 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. (C) 1997 Elsevier Science Inc.