Iterative methods for ill-conditioned Toeplitz matrices

In this paper we study the use of the Sine Transform for preconditioning linear Toeplitz systems. We consider Toeplitz matrices with a real generating function that is nonnegative with only a small number of zeros. Then we can define a preconditioner of the formS _n ΛS _n whereS _n is the matrix describing the discrete Sine transform and Λ is a diagonal matrix. If we have full knowledge aboutf then we can show that the preconditioned system is of bounded condition number independly ofn. We can obtain the same result for the case that we know only the position and order of the zeros off. If we only know the matrix and its coefficientst _j , we present Sine transform preconditioners that show in many examples the same numerical behaviour.