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

Inna Roitberg, Alexander Sakhnovich

Abstract

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 block TBT-matrices and $3$-D Toeplitz matrices.

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

Abstract

The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the -D (one-dimensional) case are classical and have numerous applications. Last year, we considered the -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 block TBT-matrices and -D Toeplitz matrices.

Paper Structure

This paper contains 7 sections, 112 equations.

Table of Contents

  1. Introduction
  2. Block TBT-matrices: preliminaries
  3. Block TBT-matrices: main results
  4. The DSTU and self-adjoint subclasses of the block TBT-matrices, and $3$-D Toeplitz matrices
  5. DSTU matrices
  6. Self-adjoint block TBT-matrices
  7. $3$-D Toeplitz matrices