Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

Acta Applicandae Mathematicae

Volumn 61, Issue 1-3, 2000, Pages 207-256

  Lubinsky, D S ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

Asymptotics; Orthogonal polynomials

Indexed keywords

EID: 0038389236     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1006470603390     Document Type: Article

References (123)

1. Achieser, N. I.: Theory of Approximation (translated by C. J. Hyman), Dover, New York, 1992.
2. Ambroladze, A.: On exceptional sets of asymptotic relations for general orthogonal polynomials, J. Approx. Theory 82 (1995), 257-273.
3. Aptekarev, A. I.: Asymptotics of orthogonal polynomials in a neighbourhood of the endpoints of the interval of orthogonality, Russian Acad. Sci. Math. Sbornik 76 (1993), 35-50.
4. p norms and entropy for general orthogonal polynomials, Russian Acad. Sci. Math. Sbornik 185 (1994), 3-30.
5. Aptekarev, A. I., Dehesa, J. S. and Yanez, R. J.: Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials, J. Math. Phys. 35 (1994), 4423-4428.
6. Aptekarev, A. I., Branquinho, A. and Marcellan, F.: Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation, to appear.
7. Barrios, D., Lopez, G. and Saff, E. B.: Asymptotics of orthogonal polynomials inside the unit circle and Szego-Padé approximants, manuscript.
8. Barrios, D., Lopez, G. and Torrano, E.: Polynomials generated by a three term recurrence relation with asymptotically periodic complex coefficients, Math. USSR Sb. 189 (1995), 629-659.
9. Beckermann, B.: On a conjecture of E. A. Rakhmanov, Constr. Approx. 16 (2000), 427-448.
10. Bleher, P. and Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. 150 (1999), 185-266.
11. 4), J. Approx. Theory 98 (1999), 146-166.
12. Bonan, S. and Clark, D. S.: Estimates of the Hermite and the Freud Polynomials, J. Approx. Theory 63 (1990), 210-224.
13. Böttcher, A. and Silbermann, B.: Analysis of Toeplitz Operators, Springer/Akademie Verlag, Berlin, 1990.
14. Böttcher, A. and Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices, Springer, Berlin, 1999.
15. Bultheel, A., Gonzalez-Vera, P., Hendriksen, E. and Njastad, O.: Orthogonal Rational Functions, Cambridge, 1999.
16. Buyarov, V. S.: Logarithmic asymptotics of polynomials orthogonal on the real axis, Soviet Math. Dokl. 43 (1991), 743-746.
17. Buyarov, V. S.: Logarithmic asymptotics of polynomials orthogonal with asymmetric weights, Math. Notes Acad. Sci. USSR 50 (1992), 789-795.
18. Buyarov, V. S. and Rakhmanov, E. A.: Families of equilibrium measures in an external field on the real axis, Math. Sb. 190 (1999), 791-802.
19. Chen, Y. and Ismail, M. E. H.: Hermitean matrix ensembles and orthogonal polynomials, Stud. Appl. Math. 100 (1998), 33-52.
20. Chen, Y., Ismail, M. E. H. and Van Assche, W.: Tau-function constructions of the recurrence coefficients of orthogonal polynomials, Adv. in Appl. Math. 20 (1998), 141-168.
21. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, 1999.
22. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S. and Zhou, X.: Asymptotics for polynomials orthogonal with respect to varying exponential weights, Internat. Math. Res. Notices 16 (1997), 759-782.
23. Deift, P., Kriecherbauer, T. and McLaughlin, K.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388-475.
24. Deift, P., Kriecherbauer, T. and McLaughlin, K.: New results for the asymptotics of orthogonal polynomials and related problems via the Lax-Levermore method, Proc. Sympos. Appl. Math. 54 (1998), 87-104.
25. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S. and Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
26. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S. and Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. (to appear).
27. Erdos, P. and Turan, P.: On interpolation II, Ann. of Math. 39 (1938), 703-724.
28. Erdos, P. and Turan, P.: On interpolation III, Ann. of Math. 41 (1940), 510-553.
29. Faber, G.: Ueber nach Polynomen fortschreitende Reihen, Sitzungsber. Bayerischen Akad. Wiss., 1922, pp.157-178.
30. Fokas, A. S., Its, A. R. and Kitaev, A. V.: Isomonodromic approach in the theory of two-dimensional quantum gravity, Uspekhi Mat. Nauk 45 (1990), 135-136.
31. Fokas, A. S., Its, A. R. and Kitaev, A. V.: Discrete Painleve equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313-344.
32. Freud, G.: Orthogonal Polynomials, Akademiai Kiado/Pergamon Press, Budapest, 1971.
33. Garnett, J. B.: Bounded Analytic Functions, Academic Press, Orlando, 1981.
34. Geronimo, J. S., Smith, D. and Van Assche, W.: Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients, J. Approx. Theory 72 (1993), 141-158.
35. Geronimo, J. S. and Van Assche, W.: Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986), 251-283.
36. Geronimo, J. S. and Van Assche, W.: Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory 62 (1990), 47-69.
37. Geronimus, Ya. L.: Orthogonal Polynomials, Consultants Bureau, New York, 1961.
38. Golinskii, L.: Schur functions, Schur parameters and orthogonal polynomials on the unit circle, Z. Anal. Anwendungen 12 (1993), 457-469.
39. Golinskii, L.: Reflection coefficients for generalized Jacobi weight functions, J. Approx. Theory 78 (1994), 117-126.
40. Golinskii, L.: Uniform asymptotics for second kind ultraspherical polynomials on the unit circle, Constr. Approx. 14 (1998), 599-608.
41. Golinskii, L.: Akhiezer's orthogonal polynomials and Bernstein-Szego method for a circular arc, J. Approx. Theory 95 (1998), 229-263.
42. 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.
43. Grenander, U. and Szego, G.: Toeplitz Forms and Their Applications, Chelsea, New York, 1984.
44. Henrici, P.: Applied and Computational Complex Analysis, Vol. 3, Wiley, 1993.
45. Hernandez, M. B. and Lopez, G.: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), 216-244.
46. 2m) (p > -1,m > 0 integer), Preprint, 1992.
47. 2m), m ≥ 2, integer, Preprint, 1993.
48. Koosis, P.: The Logarithmic Integral I, Cambridge University Press, Cambridge, 1988.
49. Kriecherbauer, T. and McLaughlin, K. T.-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights, Internat. Math. Res. Notices 6 (1999), 299-333.
50. Kuijlaars, A.: Contracted zero distributions of extremal polynomials associated with slowly decreasing weights, Manuscript.
51. Kuijlaars, A.: On the finite gap ansatz in the continuum limit of the Toda lattice, Manuscript.
52. Kuijlaars, A.: Which eigenvalues are found by the Lanczos method?, Manuscript.
53. Kuijlaars, A. and Rakhmanov, E. A.: Zero distributions for discrete orthogonal polynomials, J. Comp. Appl. Math. 99 (1998), 255-274.
54. Kuijlaars, A. and Van Assche, W.: Extremal problems on discrete sets, Proc. London Math. Soc. 76 (1999), 191-221.
55. Kuijlaars, A. and Van Assche, W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999), 167-197.
56. Landkof, N. S.: Foundations of Modern Potential Theory (translated by A. P. Doohovskoy), Springer, Berlin, 1972.
57. Levin, A. L. and Lubinsky, D. S.: Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights, Constr. Approx. 8 (1992), 463-535.
58. Levin, A. L. and Lubinsky, D. S.: Christoffel functions and orthogonal polynomials for exponential weights on [-1, 1], Mem. Amer. Math. Soc. 535 111 (1994).
59. Levin, A. L. and Lubinsky, D. S.: Bounds for orthogonal polynomials for exponential weights, J. Comp. Appl. Math. 99 (1998), 475-490.
60. Levin, A. L. and Lubinsky, D. S.: Asymptotics associated with exponential weights, Internat. Math. Res. Notices 12 (1999), 673-683.
61. Levin, A. L. and Lubinsky, D. S.: Orthogonal Polynomials for Exponential Weights, to appear.
62. Lopez, G. L.: On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Math. USSR Sb. 56 (1987), 207-219.
63. Lopez, G. L.: Asymptotics of polynomials orthogonal with respect to varying measures, Constr. Approx. 5 (1989), 199-219.
64. Lopez, G. L.: Relative asymptotics for polynomials orthogonal on the real axis, Math. USSR Sb. 65 (1990), 505-529.
65. Lopez, G. L. and Rakhmanov, E. A.: Rational approximations, orthogonal polynomials, and equilibrium distributions, In: Lecture Notes in Math. 1329, Springer, Berlin, 1988, pp. 125-157.
66. Lubinsky, D. S.: Strong Asymptotics for Extremal Errors Associated with Erdos-type Weights, Pitman Research Notes 202, Longmans, Burnt Mill, Essex, 1989.
67. Lubinsky, D. S.: A survey of general orthogonal polynomials for weights on finite and infinite intervals, Acta Appl. Math. 10 (1987), 237-296.
68. Lubinsky, D. S.: The approximate approach to orthogonal polynomials for weights on (-∞, ∞), In: P. Nevai (ed.), Orthogonal Polynomials: Theory and Practice, NATO ASI Series C 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 293-310.
69. Lubinsky, D. S.: An update on orthogonal polynomials and weighted approximation on the real line, Acta Appl. Math. 33 (1993), 121-164.
70. Lubinsky, D. S., Mhaskar, H. N. and Saff, E. B.: Freud's conjecture for exponential weights, Bull. Amer. Math. Soc. 15 (1986), 217-221.
71. Lubinsky, D. S., Mhaskar, H. N. and Saff, E. B.: A proof of Freud's conjecture for exponential weights, Constr. Approx. 4 (1988), 65-83.
72. Lubinsky, D. S. and Saff, E. B.: Strong Asymptotics for Extremal Polynomials Associated with Weights on (-∞, ∞), Lecture Notes in Math. 1305, Springer, Berlin, 1988.
73. Magnus, A.: On Freud's equations for exponential weights, J. Approx. Theory 46 (1986), 65-99.
74. Mate, A., Nevai, P. and Totik, V.: Asymptotics for the zeros of orthogonal polynomials associated with infinite intervals, J. London Math. Soc. 33 (1986), 303-310.
75. Mate, A., Nevai, P. and Totik, V.: Extensions of Szego's theory of orthogonal polynomials II, Constr. Approx. 3 (1987), 51-72.
76. Mate, A., Nevai, P. and Totik, V.: Extensions of Szego's theory of orthogonal polynomials III, Constr. Approx. 3 (1987), 73-96.
77. Mate, A., Nevai, P. and Totik, V.: Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239-282.
78. Mate, A., Nevai, P. and Zaslavsky, T.: Asymptotic expansion of ratios of coefficients of orthogonal polynomials with exponential weights, Trans. Amer. Math. Soc. 287 (1985), 495-505.
79. Mhaskar, H. N.: Introduction to the Theory of Weighted Polynomial Approximation, World Scientific, 1996.
80. Mhaskar, H. N. and Saff, E. B.: Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), 204-234.
81. Mhaskar, H. N. and Saff, E. B.: Where does the sup norm of a weighted polynomial live?, Constr. Approx. 1 (1985), 71-91.
82. p norm of a weighted polynomial live?, Trans. Amer. Math. Soc. 303 (1987), 109-124.
83. Mhaskar, H. N. and Saff, E. B.: On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990), 30-38.
84. Nevai, P.: Orthogonal polynomials, Mem. Amer. Math. Soc. 213 (1979).
85. 4), SIAM J. Math. Anal. 15 (1984), 1177-1187.
86. Nevai, P.: A new class of orthogonal polynomials, Proc. Amer. Math. Soc. 91 (1984), 4099-415.
87. Nevai, P.: Geza Freud, orthogonal polynomials and Christoffel functions: A case study, J. Approx. Theory 48 (1986), 3-167.
88. Nevai, P. (ed.), Orthogonal Polynomials and Special Functions: Theory and Practice, NATO ASI Series 294, Kluwer Acad. Publ., Dordrecht, 1990.
89. Nevai, P. and Dehesa, J. S.: On asymptotic average properties of zeros of orthogonal polynomials, SIAM J. Math. Anal. 10 (1979), 1184-1192.
90. Pan, K.: On orthogonal polynomials with respect to varying measures on the unit circle, Trans. Amer. Math. Soc. 346 (1994), 331-340.
91. Pan, K.: Extensions of Szego's theory of rational functions orthogonal on the unit circle, J. Comp. Appl. Math. 62 (1995), 321-331.
92. Pastur, L. A. and Scherbina, M.: Universality of the local eigenvalue statistics for a class of unitarily invariant random matrix ensembles, J. Statist. Phys. 86 (1997), 109-147.
93. Peherstorfer, F.: On Bernstein-Szego orthogonal polynomials on several intervals, SIAM J. Math. Anal. 21 (1990), 461-482.
94. Peherstorfer, F.: Elliptic orthogonal and extremal polynomials, Proc. London Math. Soc. 70 (1995), 605-624.
95. Peherstorfer, F.: A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), 161-185.
96. Peherstorfer, F.: Explicit generalized Zolotarev polynomials with complex coefficients, Constr. Approx. 13 (1997), 261-270.
97. Peherstorfer, F.: Minimal polynomials on several intervals with respect to the maximum norm - a survey, In: A. Martinez et al. (eds), Proc. of the Workshop "Methods of Complex Analysis in Approximation Theory", Publ. de la Universidad de Almeria, 1997, pp. 137-159.
98. 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.
99. Peherstorfer, F. and Steinbauer, R.: Comparative asymptotics for perturbed orthogonal polynomials, Trans. Amer. Math. Soc. 348 (1996), 1459-1486.
100. Peherstorfer, F. and Steinbauer, R.: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients, J. Approx. Theory 88 (1997), 316-353.
101. Qiu, W.-Y. and Wong, R.: Uniform asymptotic formula for orthogonal polynomials with exponential weight, Manuscript.
102. Rakhmanov, E. A.: On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199-213.
103. Rakhmanov, E. A.: On the asymptotics of the ratio of orthogonal polynomials II, Math. USSR Sb. 46 (1983), 105-117.
104. Rakhmanov, E. A.: On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR Sb. 47 (1984), 155-193.
105. Rakhmanov, E. A.: On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying Szego's condition, Math. USSR Sb. 58 (1987), 149-167.
106. Rakhmanov, E. A.: Strong asymptotics for orthogonal polynomials associated with exponential weights on R, In: A. A. Gonchar and E. B. Saff (eds), Methods of Approximation Theory in Complex Analysis and Mathematical Physics, Nauka, Moscow, 1992, pp. 71-97.
107. Rudin, W.: Real and Complex Analysis, McGraw-Hill, 1986.
108. Saff, E. B. and Totik, V.: Logarithmic Potentials with External Fields, Springer, Berlin, 1997.
109. 6/6), J. Approx. Theory 50 (1987), 232-293.
110. Stahl, H. B. and Totik, V.: General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.
111. Stein, E. M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Singular Integrals, Princeton University Press, Princeton, 1993.
112. Szego, G.: Orthogonal Polynomials, Amer. Math. Soc. Colloquium Publications 23, Providence, 1975.
113. Totik, V.: Weighted Approximation with Varying Weights, Lecture Notes in Math. 1300, Springer, Berlin, 1994.
114. Totik, V.: Weighted polynomial approximation for convex external fields, Constr. Approx. 16 (2000), 261-281.
115. Totik, V.: Asymptotics for Christoffel functions for general measures on the real line, Manuscript.
116. Ullman, J. L.: Orthogonal polynomials associated with an infinite interval, Michigan Math. J. 27 (1980), 353-363.
117. Ullman, J. L.: Orthogonal polynomials associated with an infinite interval, In: E. W. Cheney (ed.), Approximation Theory III, Academic Press, New York, 1980, pp. 889-895.
118. Van Assche, W.: Weighted zero distribution for polynomials orthogonal on an infinite interval, SIAM J. Math. Anal. 16 (1985), 1317-1334.
119. Van Assche, W.: Asymptotics for Orthogonal Polynomials, Lecture Notes in Math. 1265, Springer, Berlin, 1987.
120. Van Assche, W.: Asymptotic properties of orthogonal polynomials from their recurrence formula II, J. Approx. Theory 52 (1998), 322-338.
121. Van Assche, W.: Norm behaviour and zero distribution for orthogonal polynomials with nonsymmetric weights, Constr. Approx. 5 (1989), 329-345.
122. Widom, H.: Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math. 3 (1969), 127-232.
123. Zygmund, A.: Trigonometric Series, Vois. I and II, Cambridge University Press, Cambridge, 1959.