ON DISTRIBUTION OF EIGEN VALUES OF TOEPLITZ MATRICES WITH HARTWIG-FISHER SYMBOL
Abstract
Toeplitz matrices with Hartwig-Fisher symbol are investigated. The extension of Szego theorem for normal Toeplitz matrices with complex-valued symbol from L? space on the unit circle is formulated and proved.
About the Authors
Aleksander A. Batalshchikov
Don State Technical University
Russian Federation
Russian Federation
Vladimir A. Stukopin
Don State Technical University
Russian Federation
Russian Federation
References
1. Grenander V. Toeplitz Forms and Their Applications / V. Grenander, G. Szego. – Berkeley: Univ. of California Press, 1958.
2. Bottcher A. Introduction to Large Truncated Toeplitz Matrices / A. Bottcher, B. Silbermann. – New York : Universitext, Springer-Verlag, 1999.
3. Kadanoff L. Spin-Spin Correlation in the Two-Dimensional Ising Model. – Il / L. Kadanoff // Nuovo Cimento. – 1966. – B 44. – P. 276–305.
4. McCoy B.M. The Two-Dimensional Ising Model / B.M. McCoy, T.T. Wu. – Harvard : Harvard University Press, 1973.
5. Dai H. Asymptotics of eigenvalues of Toeplitz matrices / H. Dai, Z. Geary, L. Kadanoff // Journal of Statistical Mechanics. – 2009. P05012.
6. Forrester P.J. Applications and generalizations of Fisher-Hartwig asymptotics / P.J. Forrester, N.E. Frankel // Journal of Mathematical Physics. – 2004. – V. 45. – P. 2003–2028.
7. Zamarashkin N.L. Raspredelenie sobstvenny`x i singulyarny`x chisel tyoplicevy`x matricz pri oslablenny`x trebovaniyax k proizvodyashhej funkcii / N.L. Zamarashkin, E.E. Ty`rty`shnikov // Mat. sbornik. – 1997. – T. 188. – # 8. – S. 83–92. – In Russian.
8. Basor E. The Fisher-Hartvig Conjecture and Toeplitz Eingevalues / E. Basor, K. Morrison // Li-near Algebra Appl. – 1994. – V. 202. – P. 129–142.
9. Hartvig R.E. Toeplitz determinants, some applications, theorems and conjectures / R.E. Hartvig, M.E. Fisher // Adv. Chem. Phys. – 1968. – V. 15. – P. 333–353.
10. Hartvig R.E. Asymptotic behavior of Toeplitz matrices and determinants / R.E. Hartvig, M.E. Fisher // Arch. Rat. Mech. Anal. – 1969. – V. 32. – P. 190–225.
11. Widom H. Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanising index / H. Widom // Operator Theory: Adv. And Appl. – 1990. – V. 48. – P. 387–421.
12. Tyrtyshnikov E.E. A unifying approach to some old and new theorems on distribution and clustering / E.E. Tyrtyshnikov // Linear algebra and its applications. – 1996. – V. 232. – P. 1–43.
13. Hoffmann A.J. The variation of the spectrum of a normal matrix / A.J. Hoffmann, H.W. Wie-landt // Duke. Math. J. – 1953. – V. 20. – P. 37–39.
14. E`dvards R. Ryady` Fur`e v sovremennom izlozhenii / R. E`dvards. – M.: Mir, 1985. – In Russian.
Review
For citations:
Batalshchikov A.A., Stukopin V.A. ON DISTRIBUTION OF EIGEN VALUES OF TOEPLITZ MATRICES WITH HARTWIG-FISHER SYMBOL. Vestnik of Don State Technical University. 2011;11(6):812-820. (In Russ.)
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