OPUC on one foot

Bulletin of the American Mathematical Society

Volumn 42, Issue 4, 2005, Pages 431-460

  Simon, Barry ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Orthogonal polynomials; Szeg o's theorem; Verblunsky coefficients

Indexed keywords

EID: 25644448290     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-05-01075-X     Document Type: Article

References (125)

1. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965; Russian original, 1961.
2. A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Arm. SSR 22 (1987), 490-503. MR0931885 (89e:30058)
3. A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Math. USSR Sb. 53 (1986), 233-260;
4. Russian original in Mat. Sb. (N.S.) 125(167) (1984), 231-258. MR0764479 (86g:35166)
5. J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119-1178. MR1682248 (2000e:05006)
6. J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the second row of a Young diagram under Plancherel measure, Geom. Funct. Anal. 10 (2000), 702-731. MR1791137 (2001m:05258a)
7. J. Baik, P. Deift, K. T.-R. McLaughlin, P. Miller, and X. Zhou, Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys. 5 (2001), 1207-1250. MR1926668 (2003h:60141)
8. J. Baik and E. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), 1-65. MR1844203 (20021:05119)
9. G. Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471-487. MR0126126 (23:A3422)
10. G. Baxter, A norm inequality for a "finite-section" Wiener-Hopf equation, Illinois J. Math. 7 (1963), 97-103. MR0145285 (26:2818)
11. M. Bello Hernández and G. López Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), 216-244. MR1604927 (99c:42041)
12. S. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Kharkow 4 (1930), 79-93.
13. A. Borodin and E. Strahov, Averages of characteristic polynomials in random matrix theory, preprint.
14. O. Bourget, J. S. Howland, and A. Joye, Spectral analysis of unitary band matrices, Comm. Math. Phys. 234 (2003), 191-227. MR1962460 (2004c:47063)
15. 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. MR1955452 (2003k:42046)
16. C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95-115.
17. E. B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 61-82.
18. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London, 1955. MR0069338 (16:1022b)
19. D. Damanik and R. Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), 31-72.
20. G. Darboux, Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, Liouville J. (3) 4 (1878), 5-56; 377-416.
21. P. Deift and J. Ostensson, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, in preparation.
22. S. A. Denisov, On Rakhmanov's theorem for Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), 847-852. MR2019964
23. J. Dombrowski, Quasitriangular matrices, Proc. Amer. Math. Soc. 69 (1978), 95-96. MR0467373 (57:7232)
24. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties; Uspekhi Mat. Nauk 31 (1976), no. 1(187), 55-136 [Russian]. MR0427869 (55:899)
25. T. Erdélyi, P. Nevai, J. Zhang, and J. Geronimo, A simple proof of "Favard's theorem" on the unit circle, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 551-556. MR1150798 (93a:42010)
26. L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Ann. 85 (1922), 41-48.
27. H. Flaschka and D. W. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (1976), 438-456. MR0403368 (53:7179)
28. G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971.
29. I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253-304;
30. Russian original in Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309-360. MR0073805 (17:489c); MR0045281 (13:558f)
31. J. S. Geronimo, Polynomials orthogonal on the unit circle with random recurrence coefficients, in "Methods of Approximation Theory in Complex Analysis and Mathematical Physics" (Leningrad, 1991), pp. 43-61
32. Lecture Notes in Math., 1550, Springer, Berlin, 1993. MR1322290 (95k:42041)
33. J. S. Geronimo and R. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations 132 (1996), 140-178. MR1418504 (98a:39002)
34. J. S. Geronimo and R. Johnson, An inverse problem associated with polynomials orthogonal on the unit circle, Comm. Math. Phys. 193 (1998), 125-150. MR1620309 (2000a:39008)
35. 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 [Russian]. MR0012715 (7:63e)
36. Ya. L. Geronimus, On the trigonometric moment problem, Ann. of Math. (2) 47 (1946), 742-761. MR0018265 (8:265d)
37. Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79 pp. MR0061706 (15:869i)
38. Ya. L. Geronimus, On some equations in finite differences and the corresponding system of orthogonal polynomials, Zap. Mat. Otdel Fiz.-Mat. Fak. i Khar'kov. Mat. Obšč 25 (1957), 87-100 [Russian].
39. Ya. L. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau, New York, 1961. MR0133643 (24:A3469)
40. F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomials on the unit circle, preprint, 2004.
41. F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, preprint, 2004.
42. B. L. Golinskii and I. A. Ibragimov, On Szegö's limit theorem, Math. USSR Izv. 5 (1971), 421-444.
43. L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), 169-186. MR1919160 (2003e:41048)
44. L. Golinskii, Orthogonal polynomials on the unit circle, Szegö difference equations and spectral theory of unitary matrices, second Doctoral thesis, Kharkov, 2003.
45. L. Golinskii and P. Nevai, Szegö difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223 (2001), 223-259. MR1864433 (2002k:42051)
46. L. Golinskii and B. Simon, Results on spectral theorem using CMV matrices in Section 4.3 of [91].
47. I. A. Ibragimov, A theorem of Gabor Szegö, Mat. Zametki 3 (1968), 693-702 [Russian]. MR0231114 (37:6669)
48. V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), 246-251 [Russian]. MR0301565 (46:723)
49. K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476. MR1737991 (2001h:60177)
50. 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. MR0976057 (90e:42027)
51. T. Kailath, A view of three decades of linear filtering theory, IEEE Trans. Inform. Theory IT-20 (1974), 146-181. MR0465437 (57:5337)
52. T. Kailath, Signal processing applications of some moment problems, in "Moments in Mathematics," (San Antonio, Tex., 1987), pp. 71-109
53. Proc. Sympos. Appl. Math., 37, American Mathematical Society, Providence, R.I., 1987. MR0921085 (89e:94001)
54. T. Kailath, Norbert Wiener and the development of mathematical engineering, in "The Legacy of Norbert Wiener: A Centennial Symposium," Proc. Sympos. Pure Math., 60, pp. 93-116, American Mathematical Society, Providence, R.I., 1997. MR1460278 (981:01023)
55. 2(double struct T sign), J. Approx. Theory 108 (2001), 161-248. MR1815919 (2002b:42032)
56. S. Khrushchev, Classification theorems for general orthogonal polynomials on the unit circle, J. Approx. Theory 116 (2002), 268-342. MR1911083 (2003e:42036)
57. R. Killip and I. Nenciu, Matrix models for circular ensembles, Internat. Math. Res. Notices, (2004) no. 50, 2665-2701.
58. A. N. Kolmogorov, Stationary sequences in Hilbert space, Bull. Univ. Moscow 2 (1941), 40 pp. [Russian]. MR0009098 (5:101c)
59. M. G. Krein, On a generalization of some investigations of G. Szegö, W.M. Smirnov, and A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 91-94. MR0013457 (7:156b)
60. M. G. Krein, On a problem of extrapolation of A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 306-309. MR0012700 (7:61e)
61. I. M. Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215-216 [Russian]. MR0510681 (80k:58055)
62. I. M. Krichever, Appendix to "Theta-functions and nonlinear equations" by B.A. Dubrovin, Russian Math. Surveys 36 (1981), 11-92 (1982);
63. Russian original in Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11-80. MR0616797 (831:35149)
64. H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. 16 (1987), 47-77. MR0866018 (88k:42010)
65. N. Levinson, The Wiener RMS (root-mean square) error criterion in filter design and prediction, J. Math. Phys. Mass. Inst. Tech. 25 (1947), 261-278. MR0019257 (8:391e)
66. A. Martinez-Finkelshtein, K. T.-R. McLaughlin, and E. B. Saff, Strong asymptotics of Szegö orthogonal polynomials with respect to an analytic weight, preprint.
67. A. Máté, P. Nevai, and V. Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), 63-69. MR0766095 (85j:42045)
68. A. Máté, P. Nevai, and V. Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239-281. MR0882423 (88d:42040)
69. H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math. 30 (1975), 217-274. MR0397076 (53:936)
70. M. Mehta, Random Matrices, second ed., Academic Press, Inc., Boston, 1991. MR1083764 (92f:82002)
71. H. N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990), 30-38. MR1074079 (92f:42030)
72. I. Nenciu, Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, to appear in Internat. Math. Res. Notices.
73. P. Nevai, Characterization of measures associated with orthogonal polynomials on the unit circle, in "Constructive Function Theory - 86 Conference" (Edmonton, Alta., 1986)
74. Rocky Mountain J. Math. 19 (1989), 293-302. MR1016182 (90m:42034a)
75. P. Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), 322-340. MR1109411 (92f:42031)
76. P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1989), 99-104. MR1018677 (90i:33021)
77. F. Peherstorfer, Orthogonal and extremal polynomials on several intervals, in "Proc. Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA)" (Granada, 1991)
78. J. Comput. Appl. Math. 48 (1993), 187-205. MR1246858 (94m:42058)
79. F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), 161-185. MR1393285 (97d:42023)
80. F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), 180-195. MR1849545 (2002g:41009)
81. F. Peherstorfer, Inverse images of polynomial mappings and polynomials orthogonal on them, in "Proc. Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (Rome, 2001)
82. J. Comput. Appl. Math. 153 (2003), 371-385. MR1985708 (2004e:30009)
83. F. Peherstorfer and R. Steinbauer, Perturbation of orthogonal polynomials on the unit circle - a survey, In "Orthogonal Polynomials on the Unit Circle: Theory and Applications" (Madrid, 1994), pp. 97-119, Univ. Carlos III Madrid, Leganés, 1994. MR1317108 (95j:42022)
84. F. Peherstorfer and R. Steinbauer, Orthogonal polynomials on arcs of the unit circle, I, J. Approx. Theory 85 (1996), 140-184. MR1385813 (97k:42052)
85. F. Peherstorfer and R. Steinbauer, Orthogonal polynomials on arcs of the unit circle, II. Orthogonal polynomials with periodic reflection coefficients, J. Approx. Theory 87 (1996), 60-102. MR1410612 (97k:42053)
86. F. Peherstorfer and R. Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients, J. Approx. Theory 88 (1997), 316-353. MR1432577 (97m:42019)
87. F. Peherstorfer and R. Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients, II. Weak asymptotics, J. Approx. Theory 105 (2000), 102-128. MR1768526 (20011:42042)
88. F. Peherstorfer and R. Steinbauer, Orthogonal polynomials on the circumference and arcs of the circumference, J. Approx. Theory 102 (2000), 96-119. MR1736047 (2001b:42035)
89. F. Peherstorfer and R. Steinbauer, Strong asymptotics of orthonormal polynomials with the aid of Green's function, SIAM J. Math. Anal. 32 (2000), 385-402. MR1781222 (2002k:33010)
90. M. Praehofer and H. Spohn, Universal distributions for growth processes in 1 + 1 dimensions and random matrices, Phys. Rev. Lett. 84 (2000), 4882-4885.
91. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199-213.
92. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105-117.
93. E. A. Rakhmanov, Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szegö condition, Math. USSR-Sb. 58 (1987), 149-167;
94. Russian original in Mat. Sb. (N.S.) 130(172) (1986), 151-169, 284. MR0854969 (88b:42033)
95. W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, New York, 1987. MR0924157 (88k:00002)
96. I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I, II, J. Reine Angew. Math. 147 (1917), 205-232;
97. I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I, II, J. Reine Angew. Math. 148 (1918), 122-145.
98. English translation in "I. Schur Methods in Operator Theory and Signal Processing" (edited by I. Gohberg), pp. 31-59;
99. pp. 66-88, Operator Theory: Advances and Applications, 18, Birkhäuser, Basel, 1986. MR0902600 (88d:00006)
100. B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. in Math. 137 (1998), 82-203. MR1627806 (2001e:47020)
101. B. Simon, The Golinskii-Ibragimov method and a theorem of Damanik-Killip, Int. Math. Res. Not. (2003), 1973-1986. MR1991180 (2004d:42045)
102. B. Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, J. Approx. Theory. 126 (2004), 198-217. MR2045539 (2005e:42078)
103. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
104. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. MR2105089
105. B. Simon, Aizenman's theorem for orthogonal polynomials on the unit circle, to appear in Const. Approx.
106. B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, to appear in J. Approx. Theory.
107. B. Simon, Meromorphic Szegö functions and asymptotic series for Verblunsky coefficients, preprint.
108. B. Simon and T. Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), 113-125. MR1017742 (91g:81018)
109. B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75-90. MR0820340 (87k:47032)
110. G. Szegö, Über Orthogonalsysteme von Polynomen, Math. Z. 4 (1919), 139-151.
111. G. Szegö, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 6 (1920), 167-202;
112. G. Szegö, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 9 (1921), 167-190.
113. G. Szegö, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), 114-139.
114. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, R.I., 1939; 3rd edition, 1967.
115. G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sem. Math. Univ. Lund 1952 (1952), Tome Supplementaire, 228-238. MR0051961 (14:553d)
116. Talmud Bavli, Tractate Shabbos, 31a; see, for example, Schottenstein Edition, Mesorah Publications, New York, 1996.
117. V. Totik, Orthogonal polynomials with ratio asymptotics, Proc. Amer. Math. Soc. 114 (1992), 491-495. MR1065095 (92e:42017)
118. P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), 45-81. MR0650253 (58:31226)
119. S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. (2) 38 (1935), 125-157.
120. S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. (2) 40 (1936), 290-320.
121. F. Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), 9-15. MR0639135 (83b:82060)
122. B. Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554-555. MR0131120 (24:A974)
123. H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997-1013. MR0209448 (35:346)
124. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, MA; John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1949. MR0031213 (11:118j)
125. Wikipedia entry on Rodney Dangerfeld: For those in the international community who don't know of this reference, see http://en.wikipedia.org/wiki/Rodney_Dangerfield.