Jackson's Theorem and Circulant Preconditioned Toeplitz Systems

Preconditioned conjugate gradient method is used to solve n-by-n Hermitian Toeplitz systems A n x = b. The preconditioner S n is the Strang's circulant preconditioner which is deened to be the circulant matrix that copies the central diagonals of A n. The convergence rate of the method depends on the spectrum of S ?1 n A n. Using Jackson's theorem in approximation theory, we prove that if A n has a positive generating function f whosèth derivative f (`) , ` 0, is Lipschitz of order 0 < 1, then the method converges superlinearly. We show moreover that the error after 2q conjugate gradient steps decreases like Q q k=2 (log 2 k=k 2(`+)).