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

For a given nonnegative integer g, a matrix An of size n is called g-Toeplitz if its entries obey the rule An=[ar−gs]r,s=0n−1. Analogously, a matrix An again of size n is called g-circulant if An=[a(r−gs)modn]r,s=0n−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 {ak} 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 {Pn}={Pn−1An}, where {Pn} 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.