ON THE STRUCTURE OF THE INVERSE TO TOEPLITZ-BLOCK TOEPLITZ MATRICES AND OF THE CORRESPONDING POLYNOMIAL 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. We consider the 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal information, which is necessary to recover the inverse matrices, and give a complete characterization of the inverse matrices. A 2-D analogue of the important Ambartsumyan and Sobolev formulas for the corresponding reflection coefficients is derived as well.