Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Refined interlacing properties for zeros of paraorthogonal polynomials on the unit circle


Authors: K. Castillo and J. Petronilho
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 15A42
DOI: https://doi.org/10.1090/proc/14011
Published electronically: February 28, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix similar to a unitary upper Hessenberg matrix with positive subdiagonal elements.


References [Enhancements On Off] (What's this?)

  • [1] Gregory Ammar, William Gragg, and Lothar Reichel, Constructing a unitary Hessenberg matrix from spectral data, Numerical linear algebra, digital signal processing and parallel algorithms (Leuven, 1988) NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 70, Springer, Berlin, 1991, pp. 385-395. MR 1150072
  • [2] Gregory S. Ammar and William B. Gragg, Schur flows for orthogonal Hessenberg matrices, Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun., vol. 3, Amer. Math. Soc., Providence, RI, 1994, pp. 27-34. MR 1297983
  • [3] G. S. Ammar, W. B. Gragg, and L. Reichel, On the eigenproblem for orthogonal matrices, In 25th IEEE Conference on Decision and Control, pages 1963-1966, Athens, Greece, 1986.
  • [4] Peter Arbenz and Gene H. Golub, On the spectral decomposition of Hermitian matrices modified by low rank perturbations with applications, SIAM J. Matrix Anal. Appl. 9 (1988), no. 1, 40-58. MR 938057
  • [5] F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
  • [6] Ilan Bar-On, Interlacing properties of tridiagonal symmetric matrices with applications to parallel computing, SIAM J. Matrix Anal. Appl. 17 (1996), no. 3, 548-562. MR 1397244
  • [7] B. Bohnhorst, Beiträge zur numerischen Behandlung des unitären Eigenwertproblems, Ph.D. thesis, Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany, 1993.
  • [8] B. Bohnhorst, A. Bunse-Gerstner, and H. Faßbender, On the perturbation theory for unitary eigenvalue problems, SIAM J. Matrix Anal. Appl. 21 (2000), no. 3, 809-824. MR 1740872
  • [9] Angelika Bunse-Gerstner and Ludwig Elsner, Schur parameter pencils for the solution of the unitary eigenproblem, Linear Algebra Appl. 154/156 (1991), 741-778. MR 1113168
  • [10] Angelika Bunse-Gerstner and Chun Yang He, On a Sturm sequence of polynomials for unitary Hessenberg matrices, SIAM J. Matrix Anal. Appl. 16 (1995), no. 4, 1043-1055. MR 1351454
  • [11] M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29-56. MR 1955452
  • [12] Kenier Castillo, Ruymán Cruz-Barroso, and Francisco Perdomo-Pío, On a spectral theorem in paraorthogonality theory, Pacific J. Math. 280 (2016), no. 2, 327-347. MR 3453975
  • [13] P. Delsarte and Y. Genin, The tridiagonal approach to Szegő's orthogonal polynomials, Toeplitz linear systems, and related interpolation problems, SIAM J. Math. Anal. 19 (1988), no. 3, 718-735. MR 937480
  • [14] P. Delsarte and Y. Genin, On the role of orthogonal polynomials on the unit circle in digital signal processing applications, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 115-133. MR 1100290
  • [15] P. Delsarte and Y. Genin, Tridiagonal approach to the algebraic environment of Toeplitz matrices. I. Basic results, SIAM J. Matrix Anal. Appl. 12 (1991), no. 2, 220-238. MR 1089157
  • [16] P. Delsarte and Y. Genin, Tridiagonal approach to the algebraic environment of Toeplitz matrices. II. Zero and eigenvalue problems, SIAM J. Matrix Anal. Appl. 12 (1991), no. 3, 432-448. MR 1102388
  • [17] J. Geronimus, On the trigonometric moment problem, Ann. of Math. (2) 47 (1946), 742-761. MR 0018265
  • [18] J. Geronimus, On polynomials orthogonal on the circle, on trigonometric moment-problem and on allied Carathéodory and Schur functions, Rec. Math. [Mat. Sbornik] N. S. 15(57) (1944), 99-130 (Russian., with English summary). MR 0012715
  • [19] Ya. L. Geronimus. Polynomials orthogonal on the unit circle and their applications. In Series and Approximation, volume 3 of Series One, pages 1-78. Amer. Math. Soc., 1962.
  • [20] G. H. Golub, Some uses of the Lanczos algorithm in numerical linear algebra, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., Univ. Coll., Dublin, 1972) Academic Press, London, 1973, pp. 173-184. MR 0359289
  • [21] Gene H. Golub and Charles F. Van Loan, Matrix computations, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. MR 3024913
  • [22] V. B. Grègg, Positive definite Toeplitz matrices, the Hessenberg process for isometric operators, and the Gauss quadrature on the unit circle, Numerical methods of linear algebra (Russian), Moskov. Gos. Univ., Moscow, 1982, pp. 16-32 (Russian). MR 873317
  • [23] W. B. Gragg, The QR algorithm for unitary Hessenberg matrices, J. Comp. Appl. Math., 16:1-8, 1986.
  • [24] William B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle, J. Comput. Appl. Math. 46 (1993), no. 1-2, 183-198. Computational complex analysis. MR 1222480
  • [25] W. B. Gragg and L. Reichel, A divide and conquer method for unitary and orthogonal eigenproblems, Numer. Math. 57 (1990), no. 8, 695-718. MR 1065519
  • [26] R. O. Hill Jr. and B. N. Parlett, Refined interlacing properties, SIAM J. Matrix Anal. Appl. 13 (1992), no. 1, 239-247. MR 1146664
  • [27] Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
  • [28] William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113-152. MR 976057
  • [29] W. Kahan, Accurate eigenvalues of a symmetric tridiagonal matrix, Tech. report CS41, Stanford University, Stanford, CA, 1966.
  • [30] Rowan Killip and Irina Nenciu, CMV: the unitary analogue of Jacobi matrices, Comm. Pure Appl. Math. 60 (2007), no. 8, 1148-1188. MR 2330626
  • [31] H. Kimura, Generalized Schwarz form and lattice-ladder realizations of digital filters, IEEE Trans. Circuits Systems, 32:1130-1139, 1985.
  • [32] Christian Mehl, Volker Mehrmann, André C. M. Ran, and Leiba Rodman, Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations, BIT 54 (2014), no. 1, 219-255. MR 3177963
  • [33] D. S. Scott, How to make the Lanczos algorithm converge slowly, Math. Comp. 33 (1979), no. 145, 239-247. MR 514821
  • [34] Denis Serre, Matrices, Graduate Texts in Mathematics, vol. 216, Springer-Verlag, New York, 2002. Theory and applications; Translated from the 2001 French original. MR 1923507
  • [35] Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089
  • [36] Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089
  • [37] Barry Simon, Aizenman's theorem for orthogonal polynomials on the unit circle, Constr. Approx. 23 (2006), no. 2, 229-240. MR 2186307
  • [38] Barry Simon, CMV matrices: five years after, J. Comput. Appl. Math. 208 (2007), no. 1, 120-154. MR 2347741
  • [39] Barry Simon, Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle, J. Math. Anal. Appl. 329 (2007), no. 1, 376-382. MR 2306808
  • [40] Barry Simon, Szegő's theorem and its descendants, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. Spectral theory for $ L^2$ perturbations of orthogonal polynomials. MR 2743058
  • [41] R. C. Thompson and P. McEnteggert, Principal submatrices. II. The upper and lower quadratic inequalities., Linear Algebra and Appl. 1 (1968), 211-243. MR 0237532
  • [42] Pál Turán, On some open problems of approximation theory, Mat. Lapok 25 (1974), no. 1-2, 21-75 (1977) (Hungarian). MR 0442540
  • [43] P. Turán, On some open problems of approximation theory, J. Approx. Theory 29 (1980), no. 1, 23-85. P. Turán memorial volume; Translated from the Hungarian by P. Szüsz. MR 595512
  • [44] David S. Watkins, Some perspectives on the eigenvalue problem, SIAM Rev. 35 (1993), no. 3, 430-471. MR 1234638
  • [45] J. H. Wilkinson, The algebraic eigenvalue problem, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1988. Oxford Science Publications. MR 950175

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A42

Retrieve articles in all journals with MSC (2010): 15A42


Additional Information

K. Castillo
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Email: kenier@mat.uc.pt

J. Petronilho
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Email: josep@mat.uc.pt

DOI: https://doi.org/10.1090/proc/14011
Keywords: Paraorthogonal polynomials on the unit circle, zeros, unitary matrices, eigenvalues, interlacing, rank one perturbations.
Received by editor(s): June 13, 2017
Received by editor(s) in revised form: November 2, 2017
Published electronically: February 28, 2018
Communicated by: Mourad Ismail
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society