Preview

Advanced Engineering Research (Rostov-on-Don)

Advanced search

ON FORMULATION OF LIMITARY SPECTRUM OF BANDED TOEPLITZ MATRICES

Abstract

A problem on the formulation of the limit set for the sequences of increasing Toeplitz matrices eigenvalues (of the limitary spectrum) with the chosen symbol is solved. The limitary spectrum of the banded Toeplitz matrices is described as a semialgebraic set, viz a solution set of some system of algebraic equations and inequalities. In the solution to the problem, the multivariate resultant technique is used. An algorithm of the limitary spectrum determination is formulated for the particular case. An example of the limitary spectrum computation for the specific symbol is considered. These results check the classical results by F. Spitzer and P. Schmidt on the description of the limitary spectrum of the non-selfadjoint Toeplitz matrices. It should be pointed out that these classical results are significantly existence theorems, and the problem on the effective computation of limitary spectrum by F. Spitzer and P. Schmidt is not considered.

About the Authors

Svetlana Andreyevna Zolotykh
Don State Technical University.
Russian Federation


Vladimir Alexeyevich Stukopin
Stukopin, Vladimir Alexeyevich, Don State Technical University.
Russian Federation


References

1. Grenander, V., Szego, G. Teoplitz Forms and Their Applications. Berkeley: Univ. of California Press, 1958, 245 p.

2. Schmidt, P., Spitzer, F. The Teoplitz matrices of an arbitrary Laurent polynomial. Math. Scand., 1960, vol. 8, pp. 15–38.

3. Ullman, J.L. A problem of Schmidt and Spitzer. Bull. Amer. Math. Soc., 1967, vol. 73, pp. 883–885.

4. Bottcher, A., Grudsky, S. Spectral properties of banded Teoplitz matrices. Philadelphia: SIAM, 2005, 422 р.

5. Bikker, P., Uteshev, A. On the Bezout Construction of the Resultant. J. Symbolic Computation, 1999, vol. 28, pp. 45–88.

6. Cox, D., Little, J., O’Shea, D. Idealy, mnogoobraziya i algoritmy. [Ideals, Varieties, and Algorithms.] Moscow: Mir, 2000, 687 p. (in Russian).

7. Widom, H. Eigenvalue distribution of nonselfadjoint Teoplitz matrices and the asymptotics of Teoplitz determinants in the case of nonvanising index. Operator Theory: Adv. And Appl., 1990, vol. 48, pp. 387–421.


Review

For citations:


Zolotykh S.A., Stukopin V.A. ON FORMULATION OF LIMITARY SPECTRUM OF BANDED TOEPLITZ MATRICES. Vestnik of Don State Technical University. 2012;12(8):5-11. (In Russ.)

Views: 319


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2687-1653 (Online)