Abstract
In a previous paper [34] we discussed the approximation of multilevel Toeplitz matrices generated by multivariate rectangular matrix-valued continuous functions \(\), with I=[−π,π], by means of multilevel trigonometric matrix spaces with unstructured s×t blocks and by a (multi) sequence of linear approximation operators \(\). Here we prove some theorems about strong and weak clustering around the unity of the eigenvalues/singular values of \(\) and of other preconditioned matrices based on linear approximation operators. These results represent a very uniform tool for dealing with the preconditioning problem for large dimensions, for a variety of situations (e.g., control theory, Markov chain problems), both in the Hermitian and non-Hermitian/non-square case, by devising preconditioners in any multilevel trigonometric linear space of matrices.
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