Computation of Convolutions and Discrete Fourier Transforms by Polynomial Transforms

Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.