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Spectral factorization of bi-infinite multi-index block Toeplitz matrices. (English summary) Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 355–380. Block lower-diagonal-upper LDU and Cholesky LL ∗ factorizations are studied for weighted Wiener classes of bi-infinite block Toeplitz matrices A = (Ai−j) i,j∈Zd, where Aj are complex k × k matrices, and ∑ i∈Zd βi‖Ai ‖ < ∞, for a fixed sequence of weights {βi} i∈Zd subject to 1 ≤ βi+j ≤ βiβj. A further generalization, also studied in the paper under review, consists of considering the bi-infinite matrices A with the additional requirement that Ai = 0 for i ̸ ∈ J, where J is a fixed subgroup of Zd. The factorizations are considered with respect to a total order in Zd. One result: If A is positive definite, satisfies the additional requirement, and the symbol of A is invertible on the maximal ideal space, then A has a Cholesky factorization, where the factor L and its inverse also satisfy the additional requirement and belong to the Wiener class with the trivial weights βi ≡ 1; in the scalar case k = 1 the factor L in fact belongs to the weighted Wiener class. Convergence of projection methods for computing the Cholesky factorization is proved and illustrated by an example. A comparison is made with Krein’s method, which is based on the additive decomposition of the logarithm of A (when applicable)

Topics:
in, Differential and Integral Equations and Complex Analysis, Kalmytsk. Gos. Univ

Year: 2010

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