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In this article various existence results for theLDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by theLDU-factorizations of finite sections are discussed. The generalization of the results to theLDU-factorization of multi-index Toeplitz matrices is outlined.

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices