On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the 1-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the 2-D case of Toeplitz-block Toeplitz (TBT) matrices, described a minimal information, which is necessary to recover the inverse matrices, and gave a complete characterisation of the inverse matrices. Now, we develop our approach for the more complicated cases of the block TBT-matrices and of the 3-D Toeplitz matrices. Some important special cases are treated as well. (C) 2020 Elsevier Inc. All rights reserved.