Skip to main content
SpringerLink
Log in

Two of My Favorite Ways of Obtaining Asymptotics for Orthogonal Polynomials

  • Chapter
Anniversary Volume on Approximation Theory and Functional Analysis

Abstract

Improvements of the continuous and discrete Liouville-Steklov method for proving asymptotic formulas for orthogonal polynomials are discussed, and a short survey of recent asymptotic results is given.

This paper is based upon research supported by the National Science Foundation under grant No. MCS-83–00882.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahiezer, N. I., Elements of the Theory of Elliptic Functions, Nauka, Moscow, 1970.

    Google Scholar 

  2. Al-Salam, W. A. — Allaway, W. R. — Askey, R. A., Sieved ultraspherical polynomials, manuscript.

    Google Scholar 

  3. Al-Salam, W. A. — Ismail, M., Orthogonal polynomials associated with the Rogers-Ramanujan continued fraction, Pacific J. Math. 104 (1983), 269–283.

    Article  Google Scholar 

  4. Askey, R. A., Orthogonal polynomials old and new, and some combinatorial connections, in “Proceedings of the 1982 Waterloo Conference in Combinatorics” (to appear).

    Google Scholar 

  5. Askey, R. A. — Ismail, M., A generalization of ultraspherical polynomials, in “Studies in Pure Mathematics”, P. Turan memorial volume, ed. by P. Erdös, Birkhäuser, Boston, 1983.

    Google Scholar 

  6. Askey, R. A. — Ismail, M., The Rogers q-ultraspherical polynomials, in “Approximation Theory III”., ed. by W. Cheney, Academic Press, New York, 1980, 175–182.

    Google Scholar 

  7. Askey, R. A. — Ismail, M., Recurrence relations, continued fractions and orthogonal polynomials, manuscript.

    Google Scholar 

  8. Askey, R. A. — Wilson, J. A., Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials, AMS Memoirs (to appear).

    Google Scholar 

  9. Barkov, G. I., Systems of polynomials orthogonal on two symmetric intervals (in Russian), Izv. Vysh. Uceb. Zaved. Matematika 4(17) (1960), 3–16.

    Google Scholar 

  10. Baxter, G., A convergence equivalence related to polynomials orthogonal on the unit circle, TAMS 99 (1961), 471–487.

    Article  Google Scholar 

  11. Blumenthal, O., Über die Entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für (math), Dissertation, Göttingen, 1898.

    Google Scholar 

  12. Bustoz, J. — Ismail, M., The associated ultraspherical polynomials and their q-analogues, Can. J. Math. 34 (1982), 718–736.

    Article  Google Scholar 

  13. Case, K. M., Orthogonal polynomials from the viewpoint of scattering theory, J. Math. Physics 15 (1974), 2166–2174.

    Article  Google Scholar 

  14. Case, K. M., Orthogonal polynomials revisited, in “Theory and Application of Special Functions”, ed. by R. A. Askey, Academic Press, New York, 1975, 289–304.

    Google Scholar 

  15. Case, K. M., Orthogonal polynomials, II, J. Math. Physics 16 (1975), 1435–1440.

    Article  Google Scholar 

  16. Chihara, T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    Google Scholar 

  17. Chihara, T. S. — Ismail, M., Orthogonal polynomials suggested by a queueing model, Advances in Appl. Math. 3 (1982), 441–462.

    Article  Google Scholar 

  18. Erdélyi, A., Asymptotic Expansions, Dover Publ., Inc., New York, 1956.

    Google Scholar 

  19. Ford, W. R., On the integration of the homogeneous linear difference equation of second order, TAMS 10 (1909), 319–336.

    Article  Google Scholar 

  20. Freud, G., Orthogonal Polynomials, Pergamon Press, New York, 1971.

    Google Scholar 

  21. Freud, G., On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Royal Irish Acad. 76 (1976), 1–6.

    Google Scholar 

  22. Freud, G., On the greatest zero of an orthogonal polynomial, manuscript, 1978.

    Google Scholar 

  23. Frey, T., On asymptotic behavior of orthogonal polynomials (in Russian), Matem. Sbornik 49 (1959), 133–180.

    Google Scholar 

  24. Geronimo, J. S., A relation between the coefficients in the recurrence formula and the spectral function for orthogonal polynomials, TAMS 260 (1980), 65–82.

    Article  Google Scholar 

  25. Geronimo, J. S. — Case, K. M., Scattering theory and polynomials orthogonal on the unit circle, J. Math. Physics 20 (1979), 299–310.

    Article  Google Scholar 

  26. Geronimo, J. S. — Case, K. M., Scattering theory and polynomials orthogonal on the real line, TAMS 258 (1980), 467–494.

    Article  Google Scholar 

  27. Geronimo, J. S. — Nevai, P., Necessary and sufficient conditions relating the coefficients in the recurrence formula to the spectral function for orthgonal polynomials, SIAM J. Math. Analysis 14 (1983), 622–637.

    Article  Google Scholar 

  28. Geronimus, Ya. L., Polynomials Orthogonal on a Circle and Interval, Pergamon Press, New York, 1960.

    Google Scholar 

  29. Geronimus, Ya. L., Polynomials orthogonal on a circle and their applications, AMS Translations, Ser. 1, 3 (1962), 1–78.

    Google Scholar 

  30. Geronimus, Ya. L., On asymptotic properties of polynomials which are orthogonal on the unit circle, and on certain properties of positive harmonic functions, AMS Translations, Ser. 1, 3 (1962), 79–106.

    Google Scholar 

  31. Geronimus, Ya. L., Orthogonal polynomials, AMS Translations, Ser. 2, 108 (1977), 37–130.

    Google Scholar 

  32. Grenander, U. — Szegö, G., Toeplitz Forms and Their Applications, University of California Press, Berkeley, 1958.

    Google Scholar 

  33. Guseinov, G. S., The determination of an infinite Jacobi _matrix from the scattering data, Soviet Math. Dokl. 17 (1976), 596–600.

    Google Scholar 

  34. Ismail, M., The zeros of basic Bessel functions, the functions I v+ax (x), and associated orthogonal polynomials, J. Math. Analysis and Appl. 86 (1982), 1–19.

    Article  Google Scholar 

  35. Ismail, M. — Mulla, F. S., On the generalized Chebyshev polynomials, SIAM J. Math. Analysis (to appear).

    Google Scholar 

  36. Ismail, M. — Wilson, J. A., Asymptotic and generating Rations for the q-Jacobi and 4Φ3 polynomials, J. Approximation Th. 36 (1982), 43–54.

    Article  Google Scholar 

  37. Lew, J. S. — Quarles, D. A., Jr., Nonnegative solutions of a nonlinear recurrence, J. Approximation Th. 38 (1983), 357–379.

    Article  Google Scholar 

  38. Magnus, A., Recurrence coefficients for orthogonal polynomials on connected and nonconnected sets, in “Padé Approximation and its Applications”, Lecture Notes in Mathematics, No. 765, Springer-Verlag, New York, 1979, 150–171.

    Google Scholar 

  39. Máté, A. — Nevai, P., Absolute continuity of measures associated with orthogonal polynomials, in “Approximation Theory, IV”, ed. by L. Schumaker, Academic Press, 1983.

    Google Scholar 

  40. Mhaskar, H. N. — Saff, E. B., Extremal problems for polynomials with exponential weights, TAMS (to appear).

    Google Scholar 

  41. Miller, K. S., Linear Difference Equations, W. A. Benjamin, Inc., New York, 1968.

    Google Scholar 

  42. Nevai, P., Orthogonal Polynomials, AMS Memoirs, vol. 213, 1979.

    Google Scholar 

  43. Nevai, P., On orthogonal polynomials, J. Approximation Th. 25 (1979), 34–37.

    Article  Google Scholar 

  44. Nevai, P., An asymptotic formula for the derivatives of orthogonal polynomials, SIAM J. Math. Analysis 10 (1979), 472–477.

    Article  Google Scholar 

  45. Nevai, P., Distribution of zeros of orthogonal polynomials, TAMS 249 (1979), 341–361.

    Article  Google Scholar 

  46. Nevai, P., Orthogonal polynomials defined by a recurrence relation, TAMS 250 (1979), 369–384.

    Article  Google Scholar 

  47. Nevai, P. — Dehesa, J. S., On asymptotic average properties of zeros of orthogonal polynomials, SIAM J. Math. Analysis 10 (1979), 1184–1192.

    Article  Google Scholar 

  48. Nevai, P., Orthogonal polynomials associated with exp(-x4), Canadian Math. Soc. Conference Proc. 3 (1983).

    Google Scholar 

  49. Nevai, P., Asymptotics for orthogonal polynomials associated with exp(-x4), SIAM J. Math. Analysis (to appear).

    Google Scholar 

  50. Poincaré, H., On linear ordinary differential and finite difference equations (in French), Amer. J. Math. 7 (1885), 203–258.

    Article  Google Scholar 

  51. Pollaczek, F., On a generalization of the Legendre polynomials (in French), C. R. Acad. Sci. Paris 228 (1949), 1363–1365.

    Google Scholar 

  52. Pollaczek, F., On a four parameter family of orthogonal polynomials (in French), C. R. Acad. Sci. Paris 230 (1950), 2254–2256.

    Google Scholar 

  53. Pollaczek, F., On a Generalization of the Jacobi Polynomials (in French), Memorial des Sc. Math, 121 (1956), Paris.

    Google Scholar 

  54. Rahmanov, E. A., On the asymptotics of the ratio of orthogonal polynomials, I, II (in Russian), Matem. Sbornik 103 (145) (1477), 237–252 and 118 (160) (1982), 104–117.

    Google Scholar 

  55. Rahmanov, E. A., On asymptotic properties of polynomials orthogonal on the real line (in Russian), Doklady Akad. Nauk USSR 261 (1981), 282–284.

    Google Scholar 

  56. Rahmanov, E. A., On asymptotic properties of polynomials orthogonal on the real line (in Russian), Matem. Sbornik 119 (161) (1982), 163–203.

    Google Scholar 

  57. Saff, E. B., Incomplete and orthogonal polynomials, manuscript.

    Google Scholar 

  58. Sheen, R., Orthogonal Polynomials Associated with exp(-x6/6), Ph.D. Dissertation (in preparation).

    Google Scholar 

  59. Shohat, J., On the asymptotic expressions for Jacobi and Legendre polynomials derived from finite difference equations, Amer. Math. Monthly 33 (1926), 354–361.

    Article  Google Scholar 

  60. Shohat, J., On a wide class of algebraic continued fractions and the corresponding Chebyshev polynomials (in French), Comptes Rendus Acad. Sci. Paris 191 (1930), 989–990.

    Google Scholar 

  61. Shohat, J., General Theory of Chebyshev’s Orthogonal Polynomials (in French), Memorial des Sc. Math 66 (1934), Paris.

    Google Scholar 

  62. Shohat, J., A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), 401–417.

    Article  Google Scholar 

  63. Szegö, G., Orthogonal Polynomials, Amer. Math. Soc, New York, 1967.

    Google Scholar 

  64. Ullman, J. L., Orthogonal polynomials associated with an infinite interval, Mich. Math. J. 27 (1980), 353–363.

    Article  Google Scholar 

  65. Ullman, J. L., On orthogonal polynomials associated with the infinite interval, in “Approximation Theory III”, ed. by E. Cheney, Academic Press, New York, 1980, 889–895.

    Google Scholar 

  66. Ullman, J. L., A generalization of Blumenthal’s theorem on the three term recursion relationship for orthogonal polynomials, manuscript.

    Google Scholar 

  67. Wilson, J. A., Hypergeometric series, recurrence relations and some new orthogonal functions, Ph.D. Dissertation, University of Wisconsin, Madison, 1978.

    Google Scholar 

  68. Wilson, J. A., Some hypergeometric orthogonal polynomials, SIAM J. Math. Analysis 11 (1980), 690–701.

    Article  Google Scholar 

  69. Wilson, J. A., Asymptotics for the 4F3 polynomials, manuscript.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Paul Nevai

Authors
  1. Paul Nevai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lehrstuhl A für Mathematik, Rhein.-Westf. Techn. Hochschule Aachen, Templergraben 55, D-5100, Aachen, Germany

    P. L. Butzer  & R. L. Stens  & 

  2. József Attila University, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    B. Sz.-Nagy

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer Basel AG

About this chapter

Cite this chapter

Nevai, P. (1984). Two of My Favorite Ways of Obtaining Asymptotics for Orthogonal Polynomials. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-5432-0_37

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5434-4

  • Online ISBN: 978-3-0348-5432-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation