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Mat. Sb., 1997, Volume 188, Number 8, Pages 83–92 (Mi msb251)  

This article is cited in 25 scientific papers (total in 25 papers)

Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function

N. L. Zamarashkin, E. E. Tyrtyshnikov

Institute of Numerical Mathematics, Russian Academy of Sciences

Abstract: A new approach to the study of asymptotic distributions of eigenvalues and singular values of Toeplitz matrices is presented; this approach involves a matrix criterion for equidistribution. Theorems of Szego type for generating functions in $L_1$ are obtained.


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English version:
Sbornik: Mathematics, 1997, 188:8, 1191–1201

Bibliographic databases:

UDC: 512.64+517.5
MSC: Primary 15A18, 15A57; Secondary 65F15
Received: 19.06.1996

Citation: N. L. Zamarashkin, E. E. Tyrtyshnikov, “Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function”, Mat. Sb., 188:8 (1997), 83–92; Sb. Math., 188:8 (1997), 1191–1201

Citation in format AMSBIB
\by N.~L.~Zamarashkin, E.~E.~Tyrtyshnikov
\paper Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the~generating function
\jour Mat. Sb.
\yr 1997
\vol 188
\issue 8
\pages 83--92
\jour Sb. Math.
\yr 1997
\vol 188
\issue 8
\pages 1191--1201

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    This publication is cited in the following articles:
    1. N. L. Zamarashkin, E. E. Tyrtyshnikov, “On the distribution of eigenvectors of Töplitz matrices with weakened requirements on the generating function”, Russian Math. Surveys, 52:6 (1997), 1333–1334  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Zamarashkin N.L., Tyrtyshnikov E.E., “Eigenvalues of products of non-Hermitian Toeplitz matrices”, Dokl. Math., 59:3 (1999), 413–414  mathscinet  zmath  isi  elib
    3. E. E. Tyrtyshnikov, “Some applications of a matrix criterion for equidistribution”, Sb. Math., 192:12 (2001), 1877–1887  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Tyrtyshnikov E.E., Zamarashkin N.L., “Toeplitz eigenvalues for Radon measures”, Linear Algebra Appl., 343/344 (2002), 345–354  crossref  mathscinet  zmath  isi
    5. Trench W.F., “Absolute equal distribution of families of finite sets”, Linear Algebra Appl., 367 (2003), 131–146  crossref  mathscinet  zmath  isi  elib
    6. Trench W.E., “Absolute equal distribution of the spectra of Hermitian matrices”, Linear Algebra Appl., 366 (2003), 417–431  crossref  mathscinet  zmath  isi  elib
    7. Böttcher A., “Schatten norms of Toeplitz matrices with Fisher-Hartwig singularities”, Electron. J. Linear Algebra, 15 (2006), 251–259  mathscinet  zmath  isi  elib
    8. Böttcher A., Grudsky S., Ramírez de Arellano E., “On the asymptotic behavior of the eigenvectors of large banded Toeplitz matrices”, Math. Nachr., 279:1-2 (2006), 121–129  crossref  mathscinet  zmath  isi
    9. Brianzi P., Di Benedetto F., Estatico C., “Improvement of space-invariant image deblurring by preconditioned landweber iterations”, SIAM J. Sci. Comput., 30:3 (2007), 1430–1458  crossref  mathscinet  isi  elib
    10. Roch S., Silbermann B., “Szegö limit theorems for operators with almost periodic diagonals”, Oper. Matrices, 1:1 (2007), 1–29  crossref  mathscinet  isi
    11. Böttcher A., Gutiérrez-Gutiérrez J., Crespo P.M., “Mass concentration in quasicommutators of Toeplitz matrices”, J. Comput. Appl. Math., 205:1 (2007), 129–148  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Böttcher A., Grudsky S., Unterberger J., “Asymptotic pseudomodes of Toeplitz matrices”, Oper. Matrices, 2:4 (2008), 525–541  crossref  mathscinet  zmath  isi
    13. Böttcher A., Grudsky S.M., Maksimenko E.A., “The Szegő and Avram-Parter theorems for general test functions”, C. R. Math. Acad. Sci. Paris, 346:13-14 (2008), 749–752  crossref  mathscinet  zmath  isi  elib
    14. Böttcher A., Grudsky S.M., Maksimenko E.A., “Pushing the envelope of the test functions in the Szegö and Avram-Parter theorems”, Linear Algebra Appl., 429:1 (2008), 346–366  crossref  mathscinet  zmath  isi  elib
    15. Böttcher A., Grudsky S.M., Maksimenko E.A., “On the asymptotics of all eigenvalues of Hermitian Toeplitz band matrices”, Dokl. Math., 80:2 (2009), 662–664  crossref  mathscinet  zmath  isi  elib
    16. Boettcher A., Grudsky S., Schwartz M., “Some Problems Concerning the Test Functions in the Szego and Avram-Parter Theorems”, Recent Advances in Operator Theory and Applications, Operator Theory Advances and Applications, 187, 2009, 81–93  crossref  mathscinet  zmath  isi
    17. Batalschikov A.A., Stukopin V.A., “O raspredelenii sobstvennykh chisel tëplitsevykh matrits s simvolom khartviga - fishera”, Vestnik donskogo gosudarstvennogo tekhnicheskogo universiteta, 11:6 (2011), 812–820  elib
    18. Akunov T.A., Biryukov D.S., Dudarenko N.A., Polyakova M.V., Ushakov A.V., “Formirovanie spektra singulyarnykh chisel kvadratnoi matritsy prostoi struktury”, Nauchno-tekhnicheskii vestnik informatsionnykh tekhnologii, mekhaniki i optiki, 2011, 53–58  elib
    19. J.M. Bogoya, A. Böttcher, S.M. Grudsky, E.A. Maximenko, “Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols”, Journal of Mathematical Analysis and Applications, 2014  crossref
    20. Dell'Acqua P., Donatelli M., Estatico C., “Preconditioners For Image Restoration By Reblurring Techniques”, J. Comput. Appl. Math., 272 (2014), 313–333  crossref  mathscinet  zmath  isi
    21. Garoni C., Serra-Capizzano S., Vassalos P., “A general tool for determining the asymptotic spectral distribution of Hermitian matrix-sequences”, Oper. Matrices, 9:3 (2015), 549–561  crossref  mathscinet  zmath  isi  scopus
    22. Bogoya J.M., Boettcher A., Grudsky S.M., Maximenk E.A., “Eigenvectors of Hermitian Toeplitz matrices with smooth simple-loop symbols”, Linear Alg. Appl., 493 (2016), 606–637  crossref  mathscinet  zmath  isi  scopus
    23. Garoni C., Serra-Capizzano S., “The theory of locally Toeplitz sequences: a review, an extension, and a few representative applications”, Bol. Soc. Mat. Mex., 22:2, SI (2016), 529–565  crossref  mathscinet  zmath  isi
    24. Garoni C., “Topological foundations of an asymptotic approximation theory for sequences of matrices with increasing size”, Linear Alg. Appl., 513 (2017), 324–341  crossref  mathscinet  zmath  isi  scopus
    25. Zhu Zh., Wakin M.B., “on the Asymptotic Equivalence of Circulant and Toeplitz Matrices”, IEEE Trans. Inf. Theory, 63:5 (2017), 2975–2992  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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