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

摘要

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-1 ~(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.