Geodesic
On asymptotics of determinants of block Toeplitz matrices in the locally sectorial case
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 76-92

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Conditions are found for a locally sectorial block Toeplitz matrix to satisfy the Szego asymptotic formulae. Conditions are also given under which the finite section method is applicable for such a matrix. 
@article{ZNSL_1986_149_a5,
     author = {I. M. Spitkovsky},
     title = {On asymptotics of determinants of block {Toeplitz} matrices in the locally sectorial case},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {76--92},
     publisher = {mathdoc},
     volume = {149},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a5/}
}
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I. M. Spitkovsky. On asymptotics of determinants of block Toeplitz matrices in the locally sectorial case. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 76-92. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a5/