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 n−1 r,s=0. Analogously, a matrix An again of size n is called g-circulant if An =  a(r−gs) mod n n−1 r,s=0. 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 problemwhich 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} = {P−1 n An}, 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. © 2011 Elsevier B.V. All rights reserved.