Geodesic
Infinite limitedly-Toeplitz extensions of Toeplitz matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 30-39

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Infinite limitedly Toeplitz extensions of a finite rectangular Toeplitz matrix are considered. Conditions necessary and sufficient for such an extension of smallest rank to be unique are given. 
@article{ZNSL_2004_309_a3,
     author = {S. N. Il'in},
     title = {Infinite {limitedly-Toeplitz} extensions of {Toeplitz} matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {30--39},
     publisher = {mathdoc},
     volume = {309},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a3/}
}
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S. N. Il'in. Infinite limitedly-Toeplitz extensions of Toeplitz matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVII, Tome 309 (2004), pp. 30-39. http://geodesic.mathdoc.fr/item/ZNSL_2004_309_a3/