Skip to main content
SpringerLink
Log in

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

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szegő, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achieser, N. I.: Theory of Approximation (translated by C. J. Hyman), Dover, New York, 1992.

    Google Scholar 

  2. Ambroladze, A.: On exceptional sets of asymptotic relations for general orthogonal polynomials, J. Approx. Theory 82 (1995), 257–273.

    Google Scholar 

  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.

    Google Scholar 

  4. Aptekarev, A. I.: Asymptotic behaviour of Lp norms and entropy for general orthogonal polynomials, Russian Acad. Sci. Math. Sbornik 185 (1994), 3–30.

    Google Scholar 

  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.

    Google Scholar 

  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 Szeg?o-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.

    Google Scholar 

  9. eckermann, B.: On a conjecture of E. A. Rakhmanov, Constr. Approx. 16 (2000), 427–448.

    Google Scholar 

  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.

    Google Scholar 

  11. Bo, R. and Wong, R.: A uniform asymptotic formula for orthogonal polynomials associated with exp._x4/, J. Approx. Theory 98 (1999), 146–166.

    Google Scholar 

  12. Bonan, S. and Clark, D. S.: Estimates of the Hermite and the Freud Polynomials, J. Approx. Theory 63 (1990), 210–224.

    Google Scholar 

  13. Böttcher, A. and Silbermann, B.: Analysis of Toeplitz Operators, Springer/Akademie Verlag, Berlin, 1990.

    Google Scholar 

  14. Böttcher, A. and Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices, Springer, Berlin, 1999.

    Google Scholar 

  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.

    Google Scholar 

  17. Buyarov, V. S.: Logarithmic asymptotics of polynomials orthogonal with asymmetric weights, Math. Notes Acad. Sci. USSR 50 (1992), 789–795.

    Google Scholar 

  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.

    Google Scholar 

  19. Chen, Y. and Ismail, M. E. H.: Hermitean matrix ensembles and orthogonal polynomials, Stud. Appl. Math. 100 (1998), 33–52.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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. Erd?os, P. and Turan, P.: On interpolation II, Ann. of Math. 39 (1938), 703–724.

    Google Scholar 

  28. Erd?os, P. and Turan, P.: On interpolation III, Ann. of Math. 41 (1940), 510–553.

    Google Scholar 

  29. Faber, G.: Ueber nach Polynomen fortschreitende Reihen, Sitzungsber. Bayerischen Akad. Wiss., 1922, pp.157–178.

    Google Scholar 

  30. Fokas, A. S., Its, A. R. and Kitaev, A. V.: Isomonodromic approach in the theory of twodimensional quantum gravity, Uspekhi Mat. Nauk 45 (1990), 135–136.

    Google Scholar 

  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.

    Google Scholar 

  32. Freud, G.: Orthogonal Polynomials, Akademiai Kiado/Pergamon Press, Budapest, 1971.

    Google Scholar 

  33. Garnett, J. B.: Bounded Analytic Functions, Academic Press, Orlando, 1981.

    Google Scholar 

  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.

    Google Scholar 

  35. Geronimo, J. S. and Van Assche, W.: Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986), 251–283.

    Google Scholar 

  36. Geronimo, J. S. and Van Assche, W.: Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory 62 (1990), 47–69.

    Google Scholar 

  37. Geronimus, Ya. L.: Orthogonal Polynomials, Consultants Bureau, New York, 1961.

    Google Scholar 

  38. Golinskii, L.: Schur functions, Schur parameters and orthogonal polynomials on the unit circle, Z. Anal. Anwendungen 12 (1993), 457–469.

    Google Scholar 

  39. Golinskii, L.: Reflection coefficients for generalized Jacobi weight functions, J. Approx. Theory 78 (1994), 117–126.

    Google Scholar 

  40. Golinskii, L.: Uniform asymptotics for second kind ultraspherical polynomials on the unit circle, Constr. Approx. 14 (1998), 599–608.

    Google Scholar 

  41. Golinskii, L.: Akhiezer's orthogonal polynomials and Bernstein-Szeg?o method for a circular arc, J. Approx. Theory 95 (1998), 229–263.

    Google Scholar 

  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.

    Google Scholar 

  43. Grenander, U. and Szeg?o, G.: Toeplitz Forms and Their Applications, Chelsea, New York, 1984.

    Google Scholar 

  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.

    Google Scholar 

  46. Jha, Shing-Wu: The second order linear differential equations satisfied by the orthogonal polynomials associated with w.x/ D jxj_exp._x2m/ (_> _1;m > 0 integer), Preprint, 1992.

  47. Jha, Shing-Wu: Asymptotics for orthogonal polynomials associated with exp._x2m/, m > 2, integer, Preprint, 1993.

  48. Koosis, P.: The Logarithmic Integral I, Cambridge University Press, Cambridge, 1988.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  54. Kuijlaars, A. and Van Assche, W.: Extremal problems on discrete sets, Proc. London Math. Soc. 76 (1999), 191–221.

    Google Scholar 

  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.

    Google Scholar 

  56. Landkof, N. S.: Foundations of Modern Potential Theory (translated by A. P. Doohovskoy), Springer, Berlin, 1972.

    Google Scholar 

  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.

    Google Scholar 

  58. Levin, A. L. and Lubinsky, D. S.: Christoffel functions and orthogonal polynomials for exponential weights on T_1; 1U, 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.

    Google Scholar 

  60. Levin, A. L. and Lubinsky, D. S.: Asymptotics associated with exponential weights, Internat. Math. Res. Notices 12 (1999), 673–683.

    Google Scholar 

  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.

    Google Scholar 

  63. Lopez, G. L.: Asymptotics of polynomials orthogonal with respect to varying measures, Constr. Approx. 5 (1989), 199–219.

    Google Scholar 

  64. Lopez, G. L.: Relative asymptotics for polynomials orthogonal on the real axis, Math. USSR Sb. 65 (1990), 505–529.

    Google Scholar 

  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.

    Google Scholar 

  66. Lubinsky, D. S.: Strong Asymptotics for Extremal Errors Associated with Erd?os-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.

    Google Scholar 

  68. Lubinsky, D. S.: The approximate approach to orthogonal polynomials for weights on _1;1/, In: P. Nevai (ed.), Orthogonal Polynomials: Theory and Practice, NATO ASI Series C 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 293–310.

    Google Scholar 

  69. Lubinsky, D. S.: An update on orthogonal polynomials and weighted approximation on the real line, Acta Appl. Math. 33 (1993), 121–164.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  72. Lubinsky, D. S. and Saff, E. B.: Strong Asymptotics for Extremal Polynomials Associated with Weights on._1;1/, Lecture Notes in Math.1305, Springer, Berlin,1988.

    Google Scholar 

  73. Magnus, A.: On Freud's equations for exponential weights, J. Approx. Theory 46 (1986), 65–99.

    Google Scholar 

  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.

    Google Scholar 

  75. Mate, A., Nevai, P. and Totik, V.: Extensions of Szeg?o's theory of orthogonal polynomials II, Constr. Approx. 3 (1987), 51–72.

    Google Scholar 

  76. Mate, A., Nevai, P. and Totik, V.: Extensions of Szeg?o's theory of orthogonal polynomials III, Constr. Approx. 3 (1987), 73–96.

    Google Scholar 

  77. Mate, A., Nevai, P. and Totik, V.: Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239–282.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  81. Mhaskar, H. N. and Saff, E. B.: Where does the sup norm of a weighted polynomial live?, Constr. Approx. 1 (1985), 71–91.

    Google Scholar 

  82. Mhaskar, H. N. and Saff, E. B.: Where does the Lp norm of a weighted polynomial live?, Trans. Amer. Math. Soc. 303 (1987), 109–124.

    Google Scholar 

  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.

    Google Scholar 

  84. Nevai, P.: Orthogonal polynomials, Mem. Amer. Math. Soc. 213 (1979).

  85. Nevai, P.: Asymptotics for orthogonal polynomials associated with exp._x4/, SIAM J. Math. Anal. 15 (1984), 1177–1187.

    Google Scholar 

  86. Nevai, P.: A new class of orthogonal polynomials, Proc. Amer. Math. Soc. 91 (1984), 409–415.

    Google Scholar 

  87. Nevai, P.: Geza Freud, orthogonal polynomials and Christoffel functions: A case study, J. Approx. Theory 48 (1986), 3–167.

    Google Scholar 

  88. Nevai, P. (ed.), Orthogonal Polynomials and Special Functions: Theory and Practice, NATO ASI Series 294, Kluwer Acad. Publ., Dordrecht, 1990.

    Google Scholar 

  89. Nevai, P. and Dehesa, J. S.: On asymptotic average properties of zeros of orthogonal polynomials, SIAM J. Math. Anal. 10 (1979), 1184–1192.

    Google Scholar 

  90. Pan, K.: On orthogonal polynomials with respect to varying measures on the unit circle, Trans. Amer. Math. Soc. 346 (1994), 331–340.

    Google Scholar 

  91. Pan, K.: Extensions of Szeg?o's theory of rational functions orthogonal on the unit circle, J. Comp. Appl. Math. 62 (1995), 321–331.

    Google Scholar 

  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.

    Google Scholar 

  93. Peherstorfer, F.: On Bernstein-Szeg?o orthogonal polynomials on several intervals, SIAM J. Math. Anal. 21 (1990), 461–482.

    Google Scholar 

  94. Peherstorfer, F.: Elliptic orthogonal and extremal polynomials, Proc. London Math. Soc. 70 (1995), 605–624.

    Google Scholar 

  95. Peherstorfer, F.: A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), 161–185.

    Google Scholar 

  96. Peherstorfer, F.: Explicit generalized Zolotarev polynomials with complex coefficients, Constr. Approx. 13 (1997), 261–270.

    Google Scholar 

  97. Peherstorfer, F.: Minimal polynomials on several intervals with respect to the maximum norm-a survey, In: A. Martinez et al. (eds), “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.

    Google Scholar 

  99. Peherstorfer, F. and Steinbauer, R.: Comparative asymptotics for perturbed orthogonal polynomials, Trans. Amer. Math. Soc. 348 (1996), 1459–1486.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  103. Rakhmanov, E. A.: On the asymptotics of the ratio of orthogonal polynomials II,Math. USSR Sb. 46 (1983), 105–117.

    Google Scholar 

  104. Rakhmanov, E. A.: On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR Sb. 47 (1984), 155–193.

    Google Scholar 

  105. Rakhmanov, E. A.: On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying Szeg?o's condition, Math. USSR Sb. 58 (1987), 149–167.

    Google Scholar 

  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.

    Google Scholar 

  109. Sheen, R. C.: Plancherel-Rotach type asymptotics for orthogonal polynomials associated with exp._x6=6/, J. Approx. Theory 50 (1987), 232–293.

    Google Scholar 

  110. Stahl, H. B. and Totik, V.: General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.

    Google Scholar 

  111. Stein, E. M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Singular Integrals, Princeton University Press, Princeton, 1993.

    Google Scholar 

  112. Szeg?o, G.: Orthogonal Polynomials, Amer. Math. Soc. Colloquium Publications 23, Providence, 1975.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  118. Van Assche, W.:Weighted zero distribution for polynomials orthogonal on an infinite interval, SIAM J. Math. Anal. 16 (1985), 1317–1334.

    Google Scholar 

  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.

    Google Scholar 

  121. Van Assche, W.: Norm behaviour and zero distribution for orthogonal polynomials with nonsymmetric weights, Constr. Approx. 5 (1989), 329–345.

    Google Scholar 

  122. Widom, H.: Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math. 3 (1969), 127–232.

    Google Scholar 

  123. Zygmund, A.: Trigonometric Series, Vols. I and II, Cambridge University Press, Cambridge, 1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Applicable Analysis and Number Theory, Mathematics Department, Witwatersrand University South Africa, Africa

    D. S. Lubinsky

Authors
  1. D. S. Lubinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lubinsky, D.S. Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities. Acta Applicandae Mathematicae 61, 207–256 (2000). https://doi.org/10.1023/A:1006470603390

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1006470603390

Search

Navigation