OPUC on one foot. (English) Zbl 1108.42005
The theory of orthogonal polynomials on the unit circle (OPUC) invented by G. Szegő in his celebrated 1920–1921 paper has drawn much attention lately, in particular, due to a major development (the discovery of the CMV matrices). Motivated by the dearth of review literature and by the opportunity to use Schrödinger operator techniques in a new setting, the author published two volumes on this issue in 2005.
In the present expository note the way to learn about the subject in less than 1,100 pages is suggested. The author starts out with the basics of the theory (the definition of OPUC, the Szegő recurrences, both in scalar and matrix forms, the Christoffel–Darboux formula, the completeness of OPUC in the corresponding \(L^2_\mu\) space on the unit circle) and then goes over to the contribution of S. Verblunsky and Ya. L. Geronimus to the subject. The focus here is made upon Verblunsky’s “one-one” theorem and the link between OPUC and contractive holomorphic functions in the unit disk (Schur functions) due to Geronimus.
The Bernstein–Szegő approximations and zeros of OPUC are discussed in Section 4. The central Section 5 provides a brief overview of the CMV matrices as a suitable matrix representation for multiplication by \(z\) in \(L^2_\mu\) in the “right” basis.
The rest of the paper concerns Szegő’s and Baxter’s theorems, Khrushchev’s formula, transfer matrices and Weyl solutions, Rakhmanov’s theorem, exponential decay of Verblunsky coefficients and periodic OPUC.
In the present expository note the way to learn about the subject in less than 1,100 pages is suggested. The author starts out with the basics of the theory (the definition of OPUC, the Szegő recurrences, both in scalar and matrix forms, the Christoffel–Darboux formula, the completeness of OPUC in the corresponding \(L^2_\mu\) space on the unit circle) and then goes over to the contribution of S. Verblunsky and Ya. L. Geronimus to the subject. The focus here is made upon Verblunsky’s “one-one” theorem and the link between OPUC and contractive holomorphic functions in the unit disk (Schur functions) due to Geronimus.
The Bernstein–Szegő approximations and zeros of OPUC are discussed in Section 4. The central Section 5 provides a brief overview of the CMV matrices as a suitable matrix representation for multiplication by \(z\) in \(L^2_\mu\) in the “right” basis.
The rest of the paper concerns Szegő’s and Baxter’s theorems, Khrushchev’s formula, transfer matrices and Weyl solutions, Rakhmanov’s theorem, exponential decay of Verblunsky coefficients and periodic OPUC.
Reviewer: Leonid Golinskii (Kharkov)
MSC:
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 30E05 | Moment problems and interpolation problems in the complex plane |
| 42A70 | Trigonometric moment problems in one variable harmonic analysis |
Keywords:
Orthogonal polynomials; Verblunsky coefficients; Szegő’s recurrences; CMV matrices; Baxter’s theoremReferences:
| [1] | N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, 1965; Russian original, 1961. · Zbl 0124.06202 |
| [2] | A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 5, 490 – 503, 515 (Russian, with English and Armenian summaries). |
| [3] | A. I. Aptekarev, Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains, Mat. Sb. (N.S.) 125(167) (1984), no. 2, 231 – 258 (Russian). |
| [4] | Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119 – 1178. · Zbl 0932.05001 |
| [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), no. 4, 702 – 731. , https://doi.org/10.1007/PL00001635 J. Baik, P. Deift, and K. Johansson, Addendum to: ”On the distribution of the length of the second row of a Young diagram under Plancherel measure”, Geom. Funct. Anal. 10 (2000), no. 6, 1606 – 1607. · Zbl 0971.05110 · doi:10.1007/PL00001664 |
| [6] | Jinho Baik, Percy Deift, Ken T.-R. McLaughlin, Peter Miller, and Xin Zhou, Optimal tail estimates for directed last passage site percolation with geometric random variables, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1207 – 1250. · Zbl 1016.15022 · doi:10.4310/ATMP.2001.v5.n6.a7 |
| [7] | Jinho Baik and Eric M. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), no. 1, 1 – 65. · Zbl 1007.05096 · doi:10.1215/S0012-7094-01-10911-3 |
| [8] | Glen Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 (1961), 471 – 487. · Zbl 0116.35705 |
| [9] | Glen Baxter, A norm inequality for a ”finite-section” Wiener-Hopf equation, Illinois J. Math. 7 (1963), 97 – 103. · Zbl 0113.09101 |
| [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), no. 2, 216 – 244. · Zbl 0897.42016 · doi:10.1006/jath.1997.3126 |
| [11] | S. Bernstein, Sur une classe de polynomes orthogonaux, Commun. Kharkow 4 (1930), 79-93. · JFM 57.1432.01 |
| [12] | A. Borodin and E. Strahov, Averages of characteristic polynomials in random matrix theory, preprint. · Zbl 1155.15304 |
| [13] | Olivier Bourget, James S. Howland, and Alain Joye, Spectral analysis of unitary band matrices, Comm. Math. Phys. 234 (2003), no. 2, 191 – 227. · Zbl 1029.47016 · doi:10.1007/s00220-002-0751-y |
| [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. · Zbl 1022.42013 · doi:10.1016/S0024-3795(02)00457-3 |
| [15] | C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95-115. · JFM 38.0448.01 |
| [16] | E. B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math. 55 (1858), 61-82. |
| [17] | Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. · Zbl 0064.33002 |
| [18] | D. Damanik and R. Killip, Half-line Schrödinger operators with no bound states, Acta Math. 193 (2004), 31-72. · Zbl 1081.47027 |
| [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. · JFM 10.0279.01 |
| [20] | P. Deift and J. Ostensson, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, in preparation. · Zbl 1099.30027 |
| [21] | Sergey A. Denisov, On Rakhmanov’s theorem for Jacobi matrices, Proc. Amer. Math. Soc. 132 (2004), no. 3, 847 – 852. · Zbl 1050.47024 |
| [22] | J. Dombrowski, Quasitriangular matrices, Proc. Amer. Math. Soc. 69 (1978), no. 1, 95 – 96. · Zbl 0379.47013 |
| [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, Uspehi Mat. Nauk 31 (1976), no. 1(187), 55 – 136 (Russian). · Zbl 0326.35011 |
| [24] | Tamás Erdélyi, Paul Nevai, John Zhang, and Jeffrey S. Geronimo, A simple proof of ”Favard’s theorem” on the unit circle, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 2, 551 – 556. · Zbl 0753.33004 |
| [25] | L. Fejér, Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen, Math. Ann. 85 (1922), 41-48. · JFM 48.1136.03 |
| [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), no. 2, 438 – 456. · Zbl 1109.35374 · doi:10.1143/PTP.55.438 |
| [27] | G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford-New York, 1971. · Zbl 0226.33014 |
| [28] | I. M. Gel\(^{\prime}\)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. I. M. Gel\(^{\prime}\)fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309 – 360 (Russian). |
| [29] | J. S. Geronimo, Polynomials orthogonal on the unit circle with random recurrence coefficients, Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991) Lecture Notes in Math., vol. 1550, Springer, Berlin, 1993, pp. 43 – 61. · Zbl 0786.42013 · doi:10.1007/BFb0117473 |
| [30] | J. S. Geronimo and R. A. Johnson, Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle, J. Differential Equations 132 (1996), no. 1, 140 – 178. · Zbl 0863.39003 · doi:10.1006/jdeq.1996.0175 |
| [31] | J. S. Geronimo and R. Johnson, An inverse problem associated with polynomials orthogonal on the unit circle, Comm. Math. Phys. 193 (1998), no. 1, 125 – 150. · Zbl 0943.42017 · doi:10.1007/s002200050321 |
| [32] | 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). · Zbl 0060.16902 |
| [33] | J. Geronimus, On the trigonometric moment problem, Ann. of Math. (2) 47 (1946), 742 – 761. · Zbl 0060.16903 · doi:10.2307/1969232 |
| [34] | Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Translation 1954 (1954), no. 104, 79. · Zbl 0056.10303 |
| [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] | L. Ya. Geronimus, Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Authorized translation from the Russian, Consultants Bureau, New York, 1961. · Zbl 0093.26503 |
| [37] | F. Gesztesy and M. Zinchenko, A Borg-type theorem associated with orthogonal polynomials on the unit circle, preprint, 2004. · Zbl 1131.47027 |
| [38] | F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, preprint, 2004. · Zbl 1118.47023 |
| [39] | B. L. Golinskii and I. A. Ibragimov, On Szego’s limit theorem, Math. USSR Izv. 5 (1971), 421-444. · Zbl 0249.42012 |
| [40] | L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), no. 3, 169 – 186. · Zbl 1017.42014 · doi:10.1023/A:1019765002077 |
| [41] | L. Golinskii, Orthogonal polynomials on the unit circle, Szego difference equations and spectral theory of unitary matrices, second Doctoral thesis, Kharkov, 2003. |
| [42] | Leonid Golinskii and Paul Nevai, Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223 (2001), no. 2, 223 – 259. · Zbl 0998.42015 · doi:10.1007/s002200100525 |
| [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 Szegő, Mat. Zametki 3 (1968), 693 – 702 (Russian). · Zbl 0186.12201 |
| [45] | V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2 – 3, 246 – 251 (Russian, with Armenian and English summaries). |
| [46] | Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437 – 476. · Zbl 0969.15008 · doi:10.1007/s002200050027 |
| [47] | 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. · Zbl 0637.30035 · doi:10.1112/blms/21.2.113 |
| [48] | Thomas Kailath, A view of three decades of linear filtering theory, IEEE Trans. Information Theory IT-20 (1974), 146 – 181. · Zbl 0307.93040 |
| [49] | Thomas Kailath, Signal processing applications of some moment problems, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 71 – 109. · Zbl 0644.94005 · doi:10.1090/psapm/037/921085 |
| [50] | Thomas Kailath, Norbert Wiener and the development of mathematical engineering, The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 60, Amer. Math. Soc., Providence, RI, 1997, pp. 93 – 116. · Zbl 0882.45001 · doi:10.1090/pspum/060/1460278 |
| [51] | Sergei Khrushchev, Schur’s algorithm, orthogonal polynomials, and convergence of Wall’s continued fractions in \?²(\?), J. Approx. Theory 108 (2001), no. 2, 161 – 248. · Zbl 1048.42024 · doi:10.1006/jath.2000.3500 |
| [52] | S. V. Khrushchev, Classification theorems for general orthogonal polynomials on the unit circle, J. Approx. Theory 116 (2002), no. 2, 268 – 342. · Zbl 0997.42013 · doi:10.1006/jath.2002.3674 |
| [53] | R. Killip and I. Nenciu, Matrix models for circular ensembles, Internat. Math. Res. Notices, (2004) no. 50, 2665-2701. · Zbl 1255.82004 |
| [54] | A. N. Kolmogoroff, Stationary sequences in Hilbert’s space, Bolletin Moskovskogo Gosudarstvenogo Universiteta. Matematika 2 (1941), 40pp (Russian). · Zbl 0063.03291 |
| [55] | M. Krein, On a generalization of some investigations of G. Szegö, V. Smirnoff and A. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N.S.) 46 (1945), 91 – 94. · Zbl 0063.03355 |
| [56] | M. Krein, On a problem of extrapolation of A. N. Kolmogoroff, C. R. (Doklady) Acad. Sci. URSS (N. S.) 46 (1945), 306 – 309. · Zbl 0063.03356 |
| [57] | Igor Moiseevich Krichever, Algebraic curves and nonlinear difference equations, Uspekhi Mat. Nauk 33 (1978), no. 4(202), 215 – 216 (Russian). · Zbl 0382.39003 |
| [58] | B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11 – 80 (Russian). With an appendix by I. M. Krichever. · Zbl 0478.58038 |
| [59] | H. J. Landau, Maximum entropy and the moment problem, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 47 – 77. · Zbl 0617.42004 |
| [60] | Norman Levinson, The Wiener RMS (root mean square) error criterion in filter design and prediction, J. Math. Phys. Mass. Inst. Tech. 25 (1947), 261 – 278. · doi:10.1002/sapm1946251261 |
| [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. · Zbl 1135.42326 |
| [62] | Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), no. 1, 63 – 69. · Zbl 0582.42012 · doi:10.1007/BF01890022 |
| [63] | Attila Máté, Paul Nevai, and Vilmos Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), no. 2, 239 – 281. · Zbl 0633.42008 · doi:10.2307/2374574 |
| [64] | H. P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30 (1975), no. 3, 217 – 274. · Zbl 0319.34024 · doi:10.1007/BF01425567 |
| [65] | Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. · Zbl 0780.60014 |
| [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), no. 1, 30 – 38. · Zbl 0721.42014 · doi:10.1016/0021-9045(90)90111-3 |
| [67] | I. Nenciu, Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, to appear in Internat. Math. Res. Notices. · Zbl 1081.37047 |
| [68] | Paul Nevai, Characterization of measures associated with orthogonal polynomials on the unit circle, Rocky Mountain J. Math. 19 (1989), no. 1, 293 – 302. Constructive Function Theory — 86 Conference (Edmonton, AB, 1986). , https://doi.org/10.1216/RMJ-1989-19-1-293 Paul Nevai, Correction to: ”Characterizations of measures associated with orthogonal polynomials on the unit circle”, Rocky Mountain J. Math. 19 (1989), no. 4, 1111. · Zbl 0688.42021 |
| [69] | Paul Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), no. 3, 322 – 340. · Zbl 0731.42020 · doi:10.1016/0021-9045(91)90095-R |
| [70] | Paul Nevai and Vilmos Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1989), no. 1-2, 99 – 104. · Zbl 0691.42020 |
| [71] | Franz Peherstorfer, Orthogonal and extremal polynomials on several intervals, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 187 – 205. · Zbl 0790.42012 · doi:10.1016/0377-0427(93)90322-3 |
| [72] | F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), no. 2, 161 – 185. · Zbl 0854.42021 · doi:10.1007/s003659900008 |
| [73] | Franz Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), no. 2, 180 – 195. · Zbl 1025.42014 · doi:10.1006/jath.2001.3571 |
| [74] | Franz Peherstorfer, Inverse images of polynomial mappings and polynomials orthogonal on them, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 371 – 385. · Zbl 1034.30002 · doi:10.1016/S0377-0427(02)00628-3 |
| [75] | Franz Peherstorfer and Robert Steinbauer, Perturbation of orthogonal polynomials on the unit circle — a survey, Orthogonal polynomials on the unit circle: theory and applications (Madrid, 1994) Univ. Carlos III Madrid, Leganés, 1994, pp. 97 – 119. · Zbl 0956.42017 |
| [76] | Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on arcs of the unit circle. I, J. Approx. Theory 85 (1996), no. 2, 140 – 184. · Zbl 0861.42014 · doi:10.1006/jath.1996.0035 |
| [77] | Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on arcs of the unit circle. II. Orthogonal polynomials with periodic reflection coefficients, J. Approx. Theory 87 (1996), no. 1, 60 – 102. · Zbl 0946.42016 · doi:10.1006/jath.1996.0092 |
| [78] | Franz Peherstorfer and Robert Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients, J. Approx. Theory 88 (1997), no. 3, 316 – 353. · Zbl 0889.42021 · doi:10.1006/jath.1996.3026 |
| [79] | Franz Peherstorfer and Robert Steinbauer, Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II. Weak asymptotics, J. Approx. Theory 105 (2000), no. 1, 102 – 128. · Zbl 0971.42013 · doi:10.1006/jath.2000.3450 |
| [80] | Franz Peherstorfer and Robert Steinbauer, Orthogonal polynomials on the circumference and arcs of the circumference, J. Approx. Theory 102 (2000), no. 1, 96 – 119. · Zbl 0956.42017 · doi:10.1006/jath.1999.3383 |
| [81] | Franz Peherstorfer and Robert Steinbauer, Strong asymptotics of orthonormal polynomials with the aid of Green’s function, SIAM J. Math. Anal. 32 (2000), no. 2, 385 – 402. · Zbl 0970.42017 · doi:10.1137/S0036141098343045 |
| [82] | M. Praehofer and H. Spohn, Universal distributions for growth processes in \(1+1\) 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. · Zbl 0401.30033 |
| [84] | E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105-117. · Zbl 0515.30030 |
| [85] | E. A. Rakhmanov, Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szegő condition, Mat. Sb. (N.S.) 130(172) (1986), no. 2, 151 – 169, 284 (Russian). |
| [86] | Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005 |
| [87] | I. Gohberg , I. Schur methods in operator theory and signal processing, Operator Theory: Advances and Applications, vol. 18, Birkhäuser Verlag, Basel, 1986. · Zbl 0583.00020 |
| [88] | Barry Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82 – 203. · Zbl 0910.44004 · doi:10.1006/aima.1998.1728 |
| [89] | Barry Simon, The Golinskii-Ibragimov method and a theorem of Damanik and Killip, Int. Math. Res. Not. 36 (2003), 1973 – 1986. · Zbl 1042.42019 · doi:10.1155/S107379280313084X |
| [90] | Barry Simon, Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line, J. Approx. Theory 126 (2004), no. 2, 198 – 217. · Zbl 1043.42017 · doi:10.1016/j.jat.2003.12.002 |
| [91] | B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI, 2005. · Zbl 1082.42020 |
| [92] | Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. 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. · Zbl 1082.42020 |
| [93] | B. Simon, Aizenman’s theorem for orthogonal polynomials on the unit circle, to appear in Const. Approx. · Zbl 1104.42014 |
| [94] | B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, to appear in J. Approx. Theory. · Zbl 1129.42012 |
| [95] | B. Simon, Meromorphic Szego functions and asymptotic series for Verblunsky coefficients, preprint. |
| [96] | Barry Simon and Thomas Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), no. 1, 113 – 125. · Zbl 0684.47010 |
| [97] | Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75 – 90. · Zbl 0609.47001 · doi:10.1002/cpa.3160390105 |
| [98] | G. Szego, Über Orthogonalsysteme von Polynomen, Math. Z. 4 (1919), 139-151. · JFM 47.0335.01 |
| [99] | G. Szego, Beiträge zur Theorie der Toeplitzschen Formen, I, II, Math. Z. 6 (1920), 167-202; 9 (1921), 167-190. · JFM 47.0391.04 |
| [100] | G. Szego, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), 114-139. · JFM 48.0378.02 |
| [101] | G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, American Mathematical Society, Providence, R.I., 1939; 3rd edition, 1967. |
| [102] | G. Szegö, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplémentaire, 228 – 238. |
| [103] | Talmud Bavli, Tractate Shabbos, 31a; see, for example, Schottenstein Edition, Mesorah Publications, New York, 1996. |
| [104] | Vilmos Totik, Orthogonal polynomials with ratio asymptotics, Proc. Amer. Math. Soc. 114 (1992), no. 2, 491 – 495. · Zbl 0741.42015 |
| [105] | Pierre van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), no. 1, 45 – 81. · Zbl 0361.15010 · doi:10.1007/BF01418827 |
| [106] | S. Verblunsky, On positive harmonic functions: A contribution to the algebra of Fourier series, Proc. London Math. Soc. (2) 38 (1935), 125-157. · JFM 60.1131.02 |
| [107] | S. Verblunsky, On positive harmonic functions (second paper), Proc. London Math. Soc. (2) 40 (1936), 290-320. · Zbl 0013.15702 |
| [108] | Franz Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), no. 1-2, 9 – 15. · doi:10.1007/BF01292646 |
| [109] | Burton Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554 – 555. · Zbl 0099.05601 |
| [110] | Harold Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997 – 1013. · Zbl 0182.09201 |
| [111] | Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass; John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1949. · Zbl 0036.09705 |
| [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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
