A note on the ( regularizing ) preconditioning of g-Toeplitz sequences via g-circulants

For a given nonnegative integer g , a matrix A n of size n is called g -Toeplitz if its entries obey the rule A n = a r - g s ] r , s = 0 n - 1 . Analogously, a matrix A n again of size n is called g -circulant if A n = a ( r - g s ) mod n ] r , s = 0 n - 1 . In a recent work we studied the asymptotic properties, in terms of spectral distribution, of both g -circulant and g -Toeplitz sequences in the case where { a k } can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain ( - π , π ) . Here we are interested in the preconditioning problem which is well understood and widely studied in the last three decades in the classical Toeplitz case, i.e., for g = 1 . In particular, we consider the generalized case with g ? 2 and the nontrivial result is that the preconditioned sequence { P n } = { P n - 1 A n } , where { P n } is the sequence of preconditioner, cannot be clustered at 1 so that the case of g = 1 is exceptional. However, while a standard preconditioning cannot be achieved, the result has a potential positive implication since there exist choices of g -circulant sequences which can be used as basic preconditioning sequences for the corresponding g -Toeplitz structures. Generalizations to the block and multilevel case are also considered, where g is a vector with nonnegative integer entries. A few numerical experiments, related to a specific application in signal restoration, are presented and critically discussed.