Jackson's theorem and circulant preconditioned Toeplitz systems

The preconditioned conjugate gradient method is used to solve n-by-n Hermitian Toeplitz systems A_nx = b. The preconditioner S_n is the Strang's circulant preconditioner which is defined 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_n^-1A_n. Using Jackson's theorem in approximation theory, we prove that if A_n has a positive generating fucntion f{hook} whose lth derivative f{hook}^(l), l ≥ 0, is Lipschitz of order 0 < α ≤ 1, then the method converges superlinearly. We show moreover that the error after 2q conjugate gradient steps decreases like Π_{k = 2}^q (log² k k^{2(l + α)}).