Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms

We give a fast algorithm for multiplication of polynomials with real-valued coefficients without resort to complex numbers and the fast Fourier transformation. The efficiency of this algorithm is compared with the multiplication algorithm based on the discrete Hartley transformation. We demonstrate that the complexity of the Hartley transformation coincides, to within a linear term, with the complexity of the Fourier transformation, but the use of the Hartley transformation leads to a more efficient multiplication algorithm.
We give analogues of these results for finite fields. In some cases, the multiplicative constants in bounds for the complexity of multiplication of polynomials and of the Fourier and Hartley transformations over finite fields are smaller than those in the case of the field of real numbers.
This research was supported by the Russian Foundation for Basic Research, grant 99–01–01175, and the Federal Special Program ‘Integration’, grant 473.