Fast transforms of toeplitz matrices Original Research Article

We consider the problem of computing elements of the product image, where A is an N × N Toeplitz matrix and T and S are matrices denoting Fourier-transform or cosine-transform matrices. We prove that it is possible to compute p elements of  in time O(p + N log N) with only O(N) auxiliary storage. [Classical application of FFT techniques need O(p + N2log N) time and O(N2) storage.] The algorithm is not restricted to square matrices, but can handle circulant or Hankel matrices also. The algorithm is especially useful if only some of the N2 elements of  have to be computed. Even if all elements have to be computed, the algorithm is faster than traditional methods. Some applications are discussed.