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

• Authors: Claudio Estatico;Eric Ngondiep;Stefano Serra-Capizzano;Debora Sesana
• Venue: Journal of Computational and Applied Mathematics
• Year: 2012

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")"m"o"d"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 (-@p,@p). 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^-^1A"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.