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OPUC on one foot
Author(s):
Barry
Simon
Journal:
Bull. Amer. Math. Soc.
42
(2005),
431-460.
MSC (2000):
Primary 42C05, 30E05, 42A70
Posted:
June 23, 2005
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Abstract:
We present an expository introduction to orthogonal polynomials on the unit circle (OPUC).
References:
- 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. MR 0931885 (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; Russian original in Mat. Sb. (N.S.) 125(167) (1984), 231-258. MR 0764479 (86g:35166)
- 4.
- 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. MR 1682248 (2000e:05006)
- 5.
- 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. MR 1791137 (2001m:05258a)
- 6.
- 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. MR 1926668 (2003h:60141)
- 7.
- J. Baik and E. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), 1-65. MR 1844203 (2002i:05119)
- 8.
- G. Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471-487. MR 0126126 (23:A3422)
- 9.
- G. Baxter, A norm inequality for a ``finite-section" Wiener-Hopf equation, Illinois J. Math. 7 (1963), 97-103. MR 0145285 (26:2818)
- 10.
- 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. MR 1604927 (99c:42041)
- 11.
- S. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Kharkow 4 (1930), 79-93.
- 12.
- A. Borodin and E. Strahov, Averages of characteristic polynomials in random matrix theory, preprint.
- 13.
- O. Bourget, J. S. Howland, and A. Joye, Spectral analysis of unitary band matrices, Comm. Math. Phys. 234 (2003), 191-227. MR 1962460 (2004c:47063)
- 14.
- 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 (2003k:42046)
- 15.
- C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95-115.
- 16.
- E. B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 61-82.
- 17.
- E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York-Toronto-London, 1955. MR 0069338 (16:1022b)
- 18.
- D. Damanik and R. Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), 31-72.
- 19.
- 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.
- 20.
- P. Deift and J. Ostensson, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, in preparation.
- 21.
- S. A. Denisov, On Rakhmanov's theorem for Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), 847-852. MR 2019964
- 22.
- J. Dombrowski, Quasitriangular matrices, Proc. Amer. Math. Soc. 69 (1978), 95-96. MR 0467373 (57:7232)
- 23.
- 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]. MR 0427869 (55:899)
- 24.
- 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. MR 1150798 (93a:42010)
- 25.
- L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Ann. 85 (1922), 41-48.
- 26.
- 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. MR 0403368 (53:7179)
- 27.
- G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971.
- 28.
- 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; Russian original in Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309-360. MR 0073805 (17:489c); MR 0045281 (13:558f)
- 29.
- 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, Lecture Notes in Math., 1550, Springer, Berlin, 1993. MR 1322290 (95k:42041)
- 30.
- 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. MR 1418504 (98a:39002)
- 31.
- J. S. Geronimo and R. Johnson, An inverse problem associated with polynomials orthogonal on the unit circle, Comm. Math. Phys. 193 (1998), 125-150. MR 1620309 (2000a:39008)
- 32.
- 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]. MR 0012715 (7:63e)
- 33.
- Ya. L. Geronimus, On the trigonometric moment problem, Ann. of Math. (2) 47 (1946), 742-761. MR 0018265 (8:265d)
- 34.
- Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Their Applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79 pp. MR 0061706 (15:869i)
- 35.
- 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. Obsc 25 (1957), 87-100 [Russian].
- 36.
- 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. MR 0133643 (24:A3469)
- 37.
- F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomials on the unit circle, preprint, 2004.
- 38.
- F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, preprint, 2004.
- 39.
- B. L. Golinskii and I. A. Ibragimov, On Szego's limit theorem, Math. USSR Izv. 5 (1971), 421-444.
- 40.
- L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), 169-186. MR 1919160 (2003e:41048)
- 41.
- L. Golinskii, Orthogonal polynomials on the unit circle, Szego difference equations and spectral theory of unitary matrices, second Doctoral thesis, Kharkov, 2003.
- 42.
- L. Golinskii and P. Nevai, Szego difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223 (2001), 223-259. MR 1864433 (2002k:42051)
- 43.
- L. Golinskii and B. Simon, Results on spectral theorem using CMV matrices in Section 4.3 of [91].
- 44.
- I. A. Ibragimov, A theorem of Gabor Szego, Mat. Zametki 3 (1968), 693-702 [Russian]. MR 0231114 (37:6669)
- 45.
- V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), 246-251 [Russian]. MR 0301565 (46:723)
- 46.
- K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476. MR 1737991 (2001h:60177)
- 47.
- 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. MR 0976057 (90e:42027)
- 48.
- T. Kailath, A view of three decades of linear filtering theory, IEEE Trans. Inform. Theory IT-20 (1974), 146-181. MR 0465437 (57:5337)
- 49.
- T. Kailath, Signal processing applications of some moment problems, in ``Moments in Mathematics," (San Antonio, Tex., 1987), pp. 71-109, Proc. Sympos. Appl. Math., 37, American Mathematical Society, Providence, R.I., 1987. MR 0921085 (89e:94001)
- 50.
- 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. MR 1460278 (98i:01023)
- 51.
- S. Khrushchev, Schur's algorithm, orthogonal polynomials, and convergence of Wall's continued fractions in
, J. Approx. Theory 108 (2001), 161-248. MR 1815919 (2002b:42032) - 52.
- S. Khrushchev, Classification theorems for general orthogonal polynomials on the unit circle, J. Approx. Theory 116 (2002), 268-342. MR 1911083 (2003e:42036)
- 53.
- R. Killip and I. Nenciu, Matrix models for circular ensembles, Internat. Math. Res. Notices, (2004) no. 50, 2665-2701.
- 54.
- A. N. Kolmogorov, Stationary sequences in Hilbert space, Bull. Univ. Moscow 2 (1941), 40 pp. [Russian]. MR 0009098 (5:101c)
- 55.
- M. G. Krein, On a generalization of some investigations of G. Szego, W.M. Smirnov, and A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 91-94. MR 0013457 (7:156b)
- 56.
- M. G. Krein, On a problem of extrapolation of A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 46 (1945), 306-309. MR 0012700 (7:61e)
- 57.
- I. M. Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215-216 [Russian]. MR 0510681 (80k:58055)
- 58.
- I. M. Krichever, Appendix to ``Theta-functions and nonlinear equations" by B.A. Dubrovin, Russian Math. Surveys 36 (1981), 11-92 (1982); Russian original in Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11-80. MR 0616797 (83i:35149)
- 59.
- H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. 16 (1987), 47-77. MR 0866018 (88k:42010)
- 60.
- 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. MR 0019257 (8:391e)
- 61.
- A. Martinez-Finkelshtein, K. T.-R. McLaughlin, and E. B. Saff, Strong asymptotics of Szego orthogonal polynomials with respect to an analytic weight, preprint.
- 62.
- 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. MR 0766095 (85j:42045)
- 63.
- A. Máté, P. Nevai, and V. Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239-281. MR 0882423 (88d:42040)
- 64.
- H. P. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math. 30 (1975), 217-274. MR 0397076 (53:936)
- 65.
- M. Mehta, Random Matrices, second ed., Academic Press, Inc., Boston, 1991. MR 1083764 (92f:82002)
- 66.
- 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. MR 1074079 (92f:42030)
- 67.
- I. Nenciu, Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, to appear in Internat. Math. Res. Notices.
- 68.
- P. Nevai, Characterization of measures associated with orthogonal polynomials on the unit circle, in ``Constructive Function Theory--86 Conference" (Edmonton, Alta., 1986), Rocky Mountain J. Math. 19 (1989), 293-302. MR 1016182 (90m:42034a)
- 69.
- P. Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), 322-340. MR 1109411 (92f:42031)
- 70.
- P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1989), 99-104. MR 1018677 (90i:33021)
- 71.
- F. Peherstorfer, Orthogonal and extremal polynomials on several intervals, in ``Proc. Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA)" (Granada, 1991), J. Comput. Appl. Math. 48 (1993), 187-205. MR 1246858 (94m:42058)
- 72.
- F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), 161-185. MR 1393285 (97d:42023)
- 73.
- F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), 180-195. MR 1849545 (2002g:41009)
- 74.
- 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), J. Comput. Appl. Math. 153 (2003), 371-385. MR 1985708 (2004e:30009)
- 75.
- 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. MR 1317108 (95j:42022)
- 76.
- F. Peherstorfer and R. Steinbauer, Orthogonal polynomials on arcs of the unit circle, I, J. Approx. Theory 85 (1996), 140-184. MR 1385813 (97k:42052)
- 77.
- 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. MR 1410612 (97k:42053)
- 78.
- 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. MR 1432577 (97m:42019)
- 79.
- 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. MR 1768526 (2001i:42042)
- 80.
- F. Peherstorfer and R. Steinbauer, Orthogonal polynomials on the circumference and arcs of the circumference, J. Approx. Theory 102 (2000), 96-119. MR 1736047 (2001b:42035)
- 81.
- 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. MR 1781222 (2002k:33010)
- 82.
- M. Praehofer and H. Spohn, Universal distributions for growth processes in
dimensions and random matrices, Phys. Rev. Lett. 84 (2000), 4882-4885. - 83.
- E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199-213.
- 84.
- E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105-117.
- 85.
- E. A. Rakhmanov, Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szego condition, Math. USSR-Sb. 58 (1987), 149-167; Russian original in Mat. Sb. (N.S.) 130(172) (1986), 151-169, 284. MR 0854969 (88b:42033)
- 86.
- W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, New York, 1987. MR 0924157 (88k:00002)
- 87.
- I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I, II, J. Reine Angew. Math. 147 (1917), 205-232; 148 (1918), 122-145. English translation in ``I. Schur Methods in Operator Theory and Signal Processing" (edited by I. Gohberg), pp. 31-59; pp. 66-88, Operator Theory: Advances and Applications, 18, Birkhäuser, Basel, 1986. MR 0902600 (88d:00006)
- 88.
- B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. in Math. 137 (1998), 82-203. MR 1627806 (2001e:47020)
- 89.
- B. Simon, The Golinskii-Ibragimov method and a theorem of Damanik-Killip, Int. Math. Res. Not. (2003), 1973-1986. MR 1991180 (2004d:42045)
- 90.
- B. Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, J. Approx. Theory. 126 (2004), 198-217. MR 2045539 (2005e:42078)
- 91.
- B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005.
- 92.
- B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. MR 2105089
- 93.
- B. Simon, Aizenman's theorem for orthogonal polynomials on the unit circle, to appear in Const. Approx.
- 94.
- B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, to appear in J. Approx. Theory.
- 95.
- B. Simon, Meromorphic Szego functions and asymptotic series for Verblunsky coefficients, preprint.
- 96.
- B. Simon and T. Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), 113-125. MR 1017742 (91g:81018)
- 97.
- 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. MR 0820340 (87k:47032)
- 98.
- G. Szego, Über Orthogonalsysteme von Polynomen, Math. Z. 4 (1919), 139-151.
- 99.
- G. Szego, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 6 (1920), 167-202; 9 (1921), 167-190.
- 100.
- G. Szego, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), 114-139.
- 101.
- G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, R.I., 1939; 3rd edition, 1967.
- 102.
- G. Szego, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund 1952 (1952), Tome Supplementaire, 228-238. MR 0051961 (14:553d)
- 103.
- Talmud Bavli, Tractate Shabbos, 31a; see, for example, Schottenstein Edition, Mesorah Publications, New York, 1996.
- 104.
- V. Totik, Orthogonal polynomials with ratio asymptotics, Proc. Amer. Math. Soc. 114 (1992), 491-495. MR 1065095 (92e:42017)
- 105.
- P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), 45-81. MR 0650253 (58:31226)
- 106.
- S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. (2) 38 (1935), 125-157.
- 107.
- S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. (2) 40 (1936), 290-320.
- 108.
- F. Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), 9-15. MR 0639135 (83b:82060)
- 109.
- B. Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554-555. MR 0131120 (24:A974)
- 110.
- H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997-1013. MR 0209448 (35:346)
- 111.
- 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. MR 0031213 (11:118j)
- 112.
- 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.
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Additional Information:
Barry
Simon
Affiliation:
Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
Email:
bsimon@caltech.edu
DOI:
10.1090/S0273-0979-05-01075-X
PII:
S 0273-0979(05)01075-X
Keywords:
Orthogonal polynomials,
Verblunsky coefficients,
Szego's theorem
Received by editor(s):
February 2, 2005,
Received by editor(s) in revised form:
April 19, 2005
Posted:
June 23, 2005
Additional Notes:
Supported in part by NSF grant DMS-0140592.
Copyright of article:
Copyright
2005,
Barry Simon
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