Measures on the unit circle and unitary truncations of unitary operators

Journal of Approximation Theory

Volumn 139, Issue 1-2, 2006, Pages 430-468

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

^a UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Band matrices; Carath odory functions; Continued fractions; Measures on the unit circle; Normal operators; Para orthogonal polynomials; Schur parameters; Truncations of an operator

Indexed keywords

EID: 33645850693     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2005.11.001     Document Type: Article

References (30)

1. Ammar G., Gragg W.B., and Reichel L. Constructing a unitary Hessenberg matrix from spectral data. In: Golub G.H., and Van Dooren P. (Eds). Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (1991), Springer, Berlin 385-396
2. Barrios D., and López G. Ratio asymptotics for polynomials orthogonal on arcs of the unit circle. Constr. Approx. 15 (1999) 1-31
3. Barrios D., López G., Martínez A., and Torrano E. On the domain of convergence and poles of J-fractions. J. Approx. Theory 93 (1998) 177-200
4. D. Barrios, G. López, A. Martínez, E. Torrano, Finite-dimensional approximations of the resolvent of an infinite band matrix and continued fractions, Mat. Sb. 190 (1999) 23-42 [in Russian]; translation in Sb. Math. 190, 501-519.
5. Barrios D., López G., and Torrano E. Location of zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coefficients. Russ. Acad. Sci. Sb. Math. 80 (1995) 309-333
6. Cantero M.J., Moral L., and Velázquez L. Measures and para-orthogonal polynomials on the unit circle. East J. Approx. 8 (2002) 447-464
7. Cantero M.J., Moral L., and Velázquez L. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362 (2003) 29-56
8. Cantero M.J., Moral L., and Velázquez L. Minimal representations of unitary operators and orthogonal polynomials on the unit circle. Linear Algebra Appl. 408 (2005) 40-65
9. Denisov S., and Simon B. Zeros of orthogonal polynomials on the real line. J. Approx. Theory 121 (2003) 357-364
10. Geronimus Ya.L. On the character of the solutions of the moment problem in case of a limit-periodic associated fraction. Izv. Akad. Nauk SSSR. Ser. Mat. 5 (1941) 203-210 (in Russian)
11. Geronimus Ya.L. On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur functions. Mat. Sb. 15 (1944) 99-130 (in Russian)
12. Geronimus Ya.L. Orthogonal Polynomials (1961), Consultants Bureau, New York
13. Golinskii L. Singular measures on the unit circle and their reflection coefficients. J. Approx. Theory 103 (2000) 61-77
14. Golinskii L. Quadrature formula and zeros of para-orthogonal polynomials on the unit circle. Acta Math. Hungar. 96 (2002) 169-186
15. Golinskii L., Nevai P., and Van Assche W. Perturbation of orthogonal polynomials on an arc of the unit circle. J. Approx. Theory 83 (1995) 392-422
16. Golinskii L., and Nevai P. Szego{combining double acute accent} difference equations, transfer matrices and orthogonal polynomials on the unit circle. Comm. Math. Phys. 223 (2001) 223-436
17. Hahn H. Reelle Funktionen: Punktfunktionen (1948), Chelsea, New York
18. Ifantis E.K., and Panagopoulos P.N. Convergence of associated continued fractions revised. Acta Appl. Math. 66 (2001) 1-24
19. Ifantis E.K., and Panagopoulos P.N. Limit points of eigenvalues of truncated tridiagonal operators. J. Comput. Appl. Math. 133 (2001) 413-422
20. Jones W.B., Njåstad O., and Thron W.J. Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. London Math. Soc. 21 (1989) 113-152
21. Khrushchev S.V. Classification theorems for general orthogonal polynomials on the unit circle. J. Approx. Theory 116 (2002) 268-342
22. Kuratowski C. Topologie. third ed. (1961), Polska Akad. Nauk, Warsaw
23. Peherstorfer F. A special class of polynomials orthogonal on the unit circle including the associated polynomials. Constr. Approx. 12 (1996) 161-185
24. Peherstorfer F., and Steinbauer R. Orthogonal polynomials on arcs of the unit circle, II. Orthogonal polynomials with periodic reflection coefficients. J. Approx. Theory 87 (1996) 60-102
25. Simon B. Orthogonal polynomials on the unit circle: new results. Internat. Math. Res. Notices 53 (2004) 2837-2880
26. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Publication, vol. 54.1, AMS, Providence, RI, 2005.
27. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Publication, vol. 54.2, AMS, Providence, RI, 2005.
28. G. Szego{combining double acute accent}, Orthogonal Polynomials, fourth ed., AMS Colloquium Publication, vol. 23, AMS, Providence, RI, 1975.
29. W.J. Thron, L-polynomials orthogonal on the unit circle, Nonlinear numerical methods and rational approximation (Wilrijk, 1987), Mathematics and its Applications, vol. 43, Reidel, Dordrecht, 1988, pp. 271-278.
30. Wall H.S. Analytic Theory of Continued Fractions (1948), Chelsea, New York