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-gs)](r.s=0) (n-1). Analogously, a matrix An again of size n is called g-circulant if A(n) = [a ((r-gs)) 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 (-pi, pi). 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.