Minimal representations of unitary operators and orthogonal polynomials on the unit circle

Linear Algebra and Its Applications

Volumn 408, Issue 1-3, 2005, Pages 40-65

  Cantero, M J ^a   Moral, L ^a   Velazquez L ^a  

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Isometric Hessenberg matrices; Orthogonal polynomials on the unit circle; Unitary band matrices

Indexed keywords

LINEAR ALGEBRA; MATHEMATICAL MODELS; MATHEMATICAL PROGRAMMING; POLYNOMIALS; PROBLEM SOLVING; THEOREM PROVING;

ISOMETRIC HESSENBERG MATRICES; ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE; UNITARY BAND MATRICES; UNITARY OPERATORS;

MATHEMATICAL OPERATORS;

EID: 24344476659     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2005.04.025     Document Type: Article

References (38)

1. N.I. Akhiezer, and M.G. Krein Some questions in the theory of moments Trans. Math. Mono. vol. 2 1962 AMS Providence, RI Kharkov, 1938
2. M. Alfaro, El operador multiplicación en la teorIÍa de polinomios ortogonales sobre la circunferencia unidad, in: Proceedings of the II Spanish-Portuguese Mathematical Conference, Madrid, 1973, Consejo Sup. Inv. Cient., Madrid, 1977, pp. 13-21.
3. O. Bourget, J.S. Howland, and A. Joye Spectral analysis of unitary band matrices Comm. Math. Phys. 234 2003 191 227
4. M.J. Cantero, Polinomios ortogonales sobre la circunferencia unidad. Modificaciones de los parámetros de Schur, Doctoral Dissertation, Universidad de Zaragoza, 1997.
5. M.J. Cantero, L. Moral, and L. Velázquez Measures and para-orthogonal polynomials on the unit circle East J. Approx. 8 2002 447 464
6. 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
7. P. Delsarte, and Y. Genin On the role of orthogonal polynomials on the unit circle in digital signal processing applications P. Nevai Orthogonal Polynomials: Theory and Practice NATO-ASI Series C vol. 294 1990 Kluwer Dordrecht 115 133
8. R.P. Feynman Forces in molecules Phys. Rev. 56 1939 340 343
9. Ya.L. Geronimus On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur functions Mat. Sb. 15 1944 99 130
10. Ya.L. Geronimus Orthogonal Polynomials 1961 Consultants Bureau New York
11. L. Golinskii Geronimus polynomials and weak convergence on a circular arc Method. Appl. Analysis 6 1999 421 436
12. L. Golinskii Singular measures on the unit circle and their reflection coefficients J. Approx. Theory 103 2000 61 77
13. L. Golinskii Operator theoretic approach to orthogonal polynomials on an arc of the unit circle Matematicheskaya fizika, analiz, geometriya 7 2000 3 34
14. L. Golinskii On the spectra of infinite Hessenberg and Jacobi matrices Matematicheskaya fizika, analiz, geometriya 7 2000 284 298
15. L. Golinskii, and P. Nevai Szegö difference equations, transfer matrices and orthogonal polynomials on the unit circle Commun. Math. Phys. 223 2001 223 436
16. L. Golinskii, P. Nevai, and W. Van Assche Perturbation of orthogonal polynomials on an arc of the unit circle J. Approx. Theory 83 1995 392 422
17. W.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) 183-198;
18. Numerical Methods of Linear Algebra, pp. 16-32, Moskov. Gos. Univ., Moskow, 1982.
19. W.B. Gragg The QR algorithm for unitary Hessenberg matrices J. Comput. Appl. Math. 16 1986 1 8
20. W.B. Gragg, and L. Reichel A Divide and Conquer method for unitary and orthogonal eigenproblems Numer. Math. 57 1990 695 718
21. H. Hellmann Einführung in die Quantenchemie 1937 Deuticke Viena
22. E.K. Ifantis A theorem concerning differentiability of eigenvectors and eigenvalues with some applications Appl. Anal. 28 1988 257 283
23. E.K. Ifantis Concavity and convexity of eigenvalues Appl. Anal. 41 1991 209 220
24. M.E.H. Ismail, and R. Zhang On the Hellmann-Feynman theorem and the variation of zeros of certain special functions Adv. Appl. Math. 9 1988 439 446
25. W.B. Jones, O. Njåstad, and W.J. Thron Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle Bull. London Math. Soc. 21 1989 113 152
26. T. Kato Perturbation of continuous spectra by trace class operators Proc. Japan Acad. 33 1957 260 264
27. T. Kato Perturbation Theory for Linear Operators 1966 Springer New York
28. F. Marcellán, and E. Godoy Orthogonal polynomials on the unit circle: distribution of zeros J. Comput. Appl. Math. 37 1991 195 208
29. M. Reed, and B. Simon Methods of Modern Mathematical Physics, I. Functional Analysis 1972 Academic Press New York
30. M. Reed, and B. Simon Methods of Modern Mathematical Physics, IV. Analysis of Operators 1978 Academic Press New York
31. M. Rosenblum Perturbation of the continuous spectrum and unitary equivalence Pacific J. Math. 7 1957 997 1010
32. B. Simon Orthogonal Polynomials on the Unit Circle AMS Colloquim Series vols. 1 and 2 2005 American Mathematical Society Providence, RI
33. M.H. Stone Linear Transformation in Hilbert Space 1932 AMS Providence, RI
34. G. Szegö, Orthogonal Polynomials, AMS Colloq. Publ., vol. 23, 4th ed., AMS, Providence, RI, 1975.
35. A.V. Teplyaev, The pure point spectrum of random polynomials orthogonal on the unit circle, Soviet Math. Dokl. 44 (1992) 407-411;
36. Dokl. Akad. Nauk SSSR 320 (1991) 49-53.
37. W.J. Thron, L-polynomials orthogonal on the unit circle, Nonlinear numerical methods and rational approximation (Wilrijk, 1987), pp. 271-278, Math. Appl., vol. 43, Reidel, Dordrecht, 1988.
38. H. Weyl Über gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkürlicher funcktionen Math. Ann. 68 1910 220 269