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Spectral properties of Toeplitz-plus-Hankel matrices

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Abstract

We study asymptotic and uniform properties of eigenvalues of a large class of real symmetric matrices that can be decomposed into the sum of a Toeplitz matrix and a Hankel matrix. In particular, we show that their properties are essentially driven by those of the Toeplitz part. A special subclass of structured matrices arising in an approximation problem is analyzed in detail.

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References

  1. R. H. Chan,The spectrum of a family of circulant preconditioned Toeplitz systems, SIAM J. Numer. Anal.26, (1989) 503–506.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Fasino,Spectral properties of Hankel matrices and numerical solutions of finite moment problems, J. Comput. Appl. Math.65, (1995) 145–155.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Fasino, G. Inglese,Numerical schemes in the study of an inverse problem for Laplace's equation, Pubblicazioni IAGA79, (1996) CNR, Firenze, (submitted).

    Google Scholar 

  4. I. Gohberg, T. Kailath, V. Olshevsky,Fast Gaussian elimination with partial pivoting for matrices with displacement structure, Math. Comp.64, (1995) 1557–1576.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Gohberg, I. Koltracht,Efficient algorithm for Toeplitz plus Hankel matrices, Integral Equations and Operator Theory12, (1989) 136–142.

    Article  MATH  MathSciNet  Google Scholar 

  6. U. Grenander, G. Szegö, Toeplitz forms and their applications, Chelsea, New York, 1984.

    MATH  Google Scholar 

  7. G. Heinig, K. Rost,On the inverses of Toeplitz-plus-Hankel matrices, Lin. Alg. Appl.106, (1988) 39–52.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Heinig, P. Jankowsky, K. Rost,Fast inversion of Toeplitz-plus-Hankel matrices, Numer. Math.52, (1988) 665–682.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. Huckle,Circulant and skewcirculant matrices for solving Toeplitz matrix problems, SIAM J. Matrix Anal. Appl.13, (1992) 767–777.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Huckle,Fast transforms for tridiagonal linear equations, BIT34, (1994) 99–112.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Kac, W. L. Murdoch, G. Szegö,On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal.2 (1953) 767–800.

    MathSciNet  Google Scholar 

  12. T. Kailath, A. H. Sayed,Displacement structure: Theory and Applications, SIAM Rev.37, (1985) 297–386.

    Article  MathSciNet  Google Scholar 

  13. T. Ku, C.-C. Kuo,Preconditioned iterative methods for solving Toeplitz-plus-Hankel systems, SIAM J. Numer. Anal.30, (1993) 824–845.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Kuipers, H. Niederreiter, Uniform distribution of sequences, Wiley, New York 1974.

    MATH  Google Scholar 

  15. G. Inglese, F. Santosa,An inverse problem in corrosion detection, Pubblicazioni IAGA75, (1995) CNR, Firenze (submitted).

    Google Scholar 

  16. Z. Nehari,On bounded bilinear forms, Ann. of Math.65, (1957) 153–162.

    Article  MathSciNet  Google Scholar 

  17. S. V. Parter On the extreme eigenvalues of truncated Toeplitz matrices, Bull. Amer. Math. Soc.67, (1961) 191–196.

    Article  MATH  MathSciNet  Google Scholar 

  18. B. N. Parlett, The symmetric eigenvalue problem, Prentice-Hall, Englewood Cliffs 1980.

    MATH  Google Scholar 

  19. S. Serra On the extreme spectral properties of Toeplitz matrices generated by L 1 functions with several maxima/minima, BIT36, (1996) 135–142.

    Article  MATH  MathSciNet  Google Scholar 

  20. S. Serra,On the extreme eigenvalues of hermitian (block) Toeplitz matrices, preprint, (1995) to appear in: Lin. Alg. Appl.

  21. E. Tyrtyshnikov,How bad are Hankel matrices?, Numer. Math.67, (1994) 261–269.

    Article  MATH  MathSciNet  Google Scholar 

  22. E. Tyrtyshnikov,Circulant preconditioners with unbounded inverses, Lin. Alg. Appl.216, (1995) 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  23. E. Tyrtyshnikov,A unifying approach to some old and new theorems on distribution and clustering, Lin. Alg. Appl.232, (1996) 1–43.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. F. Weinberger, Variational methods for eigenvalue approximation, SIAM, Philadelphia 1974.

    MATH  Google Scholar 

  25. H. Widom,Hankel matrices, Proc. AMS100, (1965) 1–35.

    Google Scholar 

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Fasino, D. Spectral properties of Toeplitz-plus-Hankel matrices. Calcolo 33, 87–98 (1996). https://doi.org/10.1007/BF02575710

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