Fast polynomial multiplication and convolutions related to the discrete cosine transform Original Research Article

A classical scheme for multiplying polynomials is given by the Cauchy product formula. Faster methods for computing this product have been developed using circular convolution and fast Fourier transform algorithms. From the numerical point of view the Chebyshev expansion of polynomials is preferred to the monomial form. We develop a direct scheme for multiplication of polynomials in Chebyshev form as well as a fast algorithm using discrete cosine transforms. This approach leads to a new convolution operation and a new type of circulant matrices, both related to the discrete cosine transform. Extensions to bivariate polynomial products are also discussed.