FAST TRANSFORMS OF TOEPLITZ MATRICES

We consider the problem of computing elements of the product (A) over cap = TAS(T) where A is an N X 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 (A) over cap in time O(p + N log N) with only O(N) auxiliary storage. [Classical application of FFT techniques need O(p + N-2 log N) time and O(N-2) 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 N-2 elements of (A) over cap have to be computed. Even if all elements have to be computed, the algorithm is faster than traditional methods. Some applications are discussed.