  In this note, we study the fast solution of Toeplitz linear systems with
coefficient matrix $T_n(f)$, where the generating function $f$ is nonnegative
and has a unique zero at zero of any real positive order $\theta$. As
preconditioner we choose a matrix ${\tau}_n(f)$ belonging to the so-called
$\tau$ algebra, which is diagonalized by the sine transform associated to the
discrete Laplacian. In previous works, the spectral equivalence of the matrix
sequences $\{{\tau}_n(f)\}_n $ and $\{T_n(f) \}_n$ was proven under the
assumption that the order of the zero is equal to $2$: in other words the
preconditioned matrix sequence $\{{\tau}^{-1}_n(f)T_n(f) \}_n $ has
eigenvalues, which are uniformly away from zero and from infinity. Here we
prove a generalization of the above result when $\theta<2$. Furthermore, by
making use of multiple step preconditioning, we show that the matrix sequences
$\{{\tau}_n(f)\}_n $ and $\{T_n(f) \}_n$ are essentially spectrally equivalent
for every $\theta>2$, i.e., for every $\theta>2$, there exist $m_\theta$ and a
positive interval $[\alpha_\theta,\beta_\theta]$ such that all the eigenvalues
of $\{{\tau}^{-1}_n(f)T_n(f) \}_n $ belong to this interval, except at most
$m_\theta$ outliers larger than $\beta_\theta$. Such a nice property, already
known only when $\theta$ is an even positive integer greater than 2, is coupled
with the fact that the preconditioned sequence has an eigenvalue cluster at
one, so that the convergence rate of the associated preconditioned conjugate
gradient method is optimal. As a conclusion we discuss possible generalizations
and we present selected numerical experiments.
