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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients


Author: Alexander Sakhnovich
Journal: Trans. Amer. Math. Soc. 372 (2019), 5547-5570
MSC (2010): Primary 15A09, 15B05; Secondary 94A99
DOI: https://doi.org/10.1090/tran/7770
Published electronically: February 11, 2019
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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. 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.


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Additional Information

Alexander Sakhnovich
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Email: oleksandr.sakhnovych@univie.ac.at

DOI: https://doi.org/10.1090/tran/7770
Received by editor(s): August 11, 2017
Received by editor(s) in revised form: July 21, 2018, and October 8, 2018
Published electronically: February 11, 2019
Additional Notes: The author’s research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.
Article copyright: © Copyright 2019 American Mathematical Society