Skip to main content

Orthogonal polynomials

Part of the Lecture Notes in Mathematics book series (LNM,volume 1574)

Keywords

  • Orthogonal Polynomial
  • Exponential Weight
  • Linear Difference Equation
  • Compact Perturbation
  • Orthonormal Polynomial

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • [ADMR] Alfaro M., Dehesa, J. S., Marcellán, F. J., Rubio de Francia J. L., Vinuesa J., eds., Orthogonal Polynomials and Their Applications, Lecture Notes in Mathematics, Vol. 1329, Springer Verlag, Berlin, 1988.

    Google Scholar 

  • [As] Askey R., Orthogonal Polynomials and Special Functions, Regional Conference Series in Applied Mathematics, Vol. 21, SIAM, Philadelphia, 1975.

    CrossRef  MATH  Google Scholar 

  • [BN] Bonan S. S. and Nevai P., Orthogonal polynomials and their derivatives, I, J. Approx. Theory 40 (1984), 134–147.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [BLN] Bonan S. S., Lubinsky D. S. and Nevai P., Orthogonal polynomials and their derivatives, II, SIAM J. Math. Anal. 18 (1987), 1163–1176.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [BDMR] Brezinski C., Draux A., Magnus A. P. and Ronveaux A., eds., Polynômes Orthogonaux et Applications, Proceedings, Bar-le-Duc, 1984, Lecture Notes in Mathematics, Vol. 1171, Springer Verlag, Berlin, 1985.

    MATH  Google Scholar 

  • [BGR] Brezinski C., Gori L. and Ronveaux A., eds. 1991.

    Google Scholar 

  • [CT] Chihara T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York-London-Paris, 1978.

    MATH  Google Scholar 

  • [CR] Chin R. C. Y., A domain decomposition method for generating orthogonal polynomials for a Gaussian weight on a finite interval, J. Comp. Phys. 99 (1992), 321–336.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Ga] Gautschi W., Computational aspects of orthogonal polynomials, Orthogonal Polynomials: Theory and Practice, NATO ASI Series C: Mathematical and Physical Scieces, Vol. 294, P. Nevai, ed., Kluwer Academic Publishers, Dordrecht-Boston-London, 1990, pp. 181–216.

    CrossRef  Google Scholar 

  • [GVA] Geronimo J. S. and Van Assche W., Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991), 341–371.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Lo] López G. L., Convergence of Padé approximants of Stieltjes type meriomorphic functions and comparative asymptotics for orthogonal polynomials, Math. USSR Sb. 64 (1989), 207–227.

    CrossRef  MathSciNet  Google Scholar 

  • [Lu1] Lubinsky D. S., A survey of general orthogonal polynomials for weights on finite and infinite intervals, Acta Appl. Math. 10 (1987), 237–296.

    MathSciNet  MATH  Google Scholar 

  • [Lu2] Lubinsky D. S., Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdös-type Weights, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Vol. 202, Harlow, United Kingdom, 1988.

    Google Scholar 

  • [LN] Lubinsky D. S. and Nevai P., Sub-exponential growth of solutions of difference equations, J. London Math. Soc. (to appear).

    Google Scholar 

  • [LS] Lubinsky D. S. and Saff E. B., Strong asymptotics for extremal polynomials associated with weights on ℝ, Lecture Notes in Mathematics, vol. 1305, Springer-Verlag, Berlin, 1988.

    CrossRef  MATH  Google Scholar 

  • [Ma1] Magnus A. P., On Freud’s equations for exponential weights, J. Approx. Theory 46 (1986), 65–99.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Ma2] Magnus A. P., Toeplitz matrix techniques and convergence of complex weight Padé approximants, J. Comput. Appl. Math. 19 (1987), 23–38.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [MN1] Máté A. and Nevai P., Orthogonal polynomials and absolutely continuous measures, Approximation Theory IV, Chui C. K., Schumaker, L. L., and Ward J. D., eds., Academic Press, New York, 1983, pp. 611–617.

    Google Scholar 

  • [MN2] Máté A. and Nevai P., Eigenvalues of finite band-width Hilbert space operators and their application to orthogonal polynomials, Can. J. Math. 16 (1989), 106–122.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [MNT1] Máté A., Nevai P. and Totik V., Asymptotics for the leading coefficients of orthogonal polynomials on the unit circle, Constr. Approx. 1 (1985), 63–69.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [MNT2] Máté A., Nevai P. and Totik V., Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1 (1985), 231–248.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [MNT3] Máté A., Nevai P. and Totik V., Extensions of Szegő’s theory of orthogonal polynomials, II & III, Constr. Approx. 3 (1987), 51–72 & 73–96.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [MNT4] Máté A., Nevai P. and Totik V., Szegő’s extremum problem on the unit circle, Ann. Math. 134 (1991), 433–453.

    CrossRef  MATH  Google Scholar 

  • [Mi] Miller K. S., Linear Difference Equations, W. A. Benjamin, Inc., New York, 1968.

    MATH  Google Scholar 

  • [N1] Nevai P., Orthogonal Polynomials, vol. 213, Mem. Amer. Math. Soc., Providence, Rhode Island, 1979.

    MATH  Google Scholar 

  • [N2] Nevai P., Orthogonal polynomials associated with exp(−x4), Canadian Math. Soc. Conf. Proc. 3 (1983), 263–285.

    MathSciNet  Google Scholar 

  • [N3] Nevai P., Géza Freud, orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory 48, (1986), 3–167.

    CrossRef  MATH  Google Scholar 

  • [N4] Nevai P., Research problems in orthogonal polynomials, Approximation Theory VI, Vol. II, Chui C. K., Schumaker L. L. and Ward J. D., eds., Academic Press, Inc., Boston, 1989, pp. 449–489.

    Google Scholar 

  • [N5] Nevai P., ed., Orthogonal Polynomials: Theory and Practice, NATO ASI Series C: Mathematical and Physical Sciences, Vol. 294, Kluwer Academic Publishers, Dordrecht-Boston-London, 1990.

    MATH  Google Scholar 

  • [N6] Nevai P., Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991), 322–340.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [NTZ] Nevai P., Totik V. and Zhang J., Orthogonal polynomials: their growth relative to their sums, J. Approx. Theory 67 (1991), 215–234.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [Ra] Rahmanov E. A., On the asymptotics of the ratio of orthogonal polynomials, Math. USSR-Sb. 33 (1977), 199–213; II, ibid Math. USSR-Sb. 46 (1983), 105–117.

    CrossRef  MathSciNet  Google Scholar 

  • [SaT] Saff E. B. and Totik V., Logarithmic Potentials with External Fields, in preparation.

    Google Scholar 

  • [StT] Stahl H. and Totik V., General Orthogonal Polynomials, Encyclopedia of Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1992.

    CrossRef  MATH  Google Scholar 

  • [To] Totik V., Orthogonal polynomials with ratio asymptotics, Proc. Amer. Math. Soc. 114 (1992), 491–495.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • [VA1] Van Assche W., Asymptotics for orthogonal polynomials, Lectures Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987.

    CrossRef  MATH  Google Scholar 

  • [VA2] Van Assche W., Constructive methods in the analysis of orthogonal polynomials, Dissertation, Katholieke Universiteit Leuven, Heverlee (Leuven), Belgium, 1992.

    Google Scholar 

  • [Zh] Zhang J., Relative growth of linear iterations and orthogonal polynomials on several intervals, J. Lin. Alg. Appl. (to appear).

    Google Scholar 

References

  • Davis P. J., Interpolation and Approximation, Blaisdell, 1963.

    Google Scholar 

  • Gaier D., Vorlesungen über Approximation im Komplexen., Birkhäuser Verlag, 1980.

    Google Scholar 

  • Lempert L., Recursion for orthogonal polynomials on complex domains, Fourier Analysis and Approximation Theory (G. Alexis, P. Turán, eds.), Vol. II. Colloquia Mathematica Societatis János Bolyai 19, North-Holland, 1978, pp. 481–494.

    Google Scholar 

  • Stahl H., Totik V., General Orthogonal Polynomials, Encyclopedia of Mathematics, Cambridge University Press (to appear).

    Google Scholar 

References

  • Bochner S., Uber Sturm-Liouvillesche Polynomsysteme, Math. Zeit. 29 (1929), 730–736.

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Hahn W., Uber die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Zeit. 39 (1935), 634–638.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Iserles A., Koch P. E., Nørsett S. P. and Sanz-Serna J. M., On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Th. 65 (1991), 151–175.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Iserles A., Koch P. E., Nørsett S. P. Sanz-Serna J. M., Polynomials orthogonal in a Sobolev space and pseudospectral methods (1991) (to appear).

    Google Scholar 

  • Krall A. M., Orthogonal polynomials satisfying a certain fourth-order differential equations, Proc. Roy. Soc. Edinburgh 87 (1981), 271–288.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Krall H. L., On orthogonal polynomials satisfying a certain fourth-order differential equations, The Pennsylvania State College Studies (1940), no. 6.

    Google Scholar 

  • Sonine N. Ja., Uber die angenäherte Berechnung der bestimmten Integrale und über die dabei vorkommenden ganzen Functionen, Warsaw Univ. Izv. 18 (1887), 1–76. (Russian)

    MATH  Google Scholar 

References

  • Braess D., On the Conjecture of Meinardus on Rational Approximation of ex, II, J. Approx. Theory 40 (1984), 375–379.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Goncar A. A., Rahmanov E. A., Equilibrium Distributions and Degree of Rational Approximation of Analytic Functions, Math. USSR Sbornik 62 (1989).

    Google Scholar 

  • Levin A. L., Lubinsky D. S., Rows and Diagonals of the Walsh Array for Entire Functions with Smooth Maclaurin Series Coefficients, Constr. Approx. 3 (1990), 257–286.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lubinsky D. S., Distribution of Poles of Diagonal Rational Approximants to Functions of Fast Rational Approximability, Constr. Approx. 7 (1991), 501–519.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lubinsky D. S., Rational Versus Polynomial Approximation of Entire Functions, IMACS Annals on Computing and Applied Mathematics 9 (1991), 81–86.

    MathSciNet  MATH  Google Scholar 

  • Parfenov O. G., Estimates of the Singular Numbers of the Carleson Imbedding Operator, Math. USSR Sbornik 59 (1988), 497–514.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Petrushev P. P., Popov V. A., Rational Approximation of Real Functions, Encyclopaedia of Mathematics and its Applications, vol. 28, Cambridge University Press, Cambridge, 1988.

    CrossRef  MATH  Google Scholar 

  • Saff E. B., On the Degree of Best Rational Approximation to the Exponential Function, J. Approx. Theory 9 (1973), 97–101.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Stahl H., A Note on Three Conjectures by Gonchar on Rational Approximation, J. Approx. Theory 50 (1987), 3–7.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Stahl H., General Convergence Results for Padé Approximants, Approximation Theory VI (Chui C. K., et al., eds.), Academic Press, San Diego, 1989, pp. 605–634.

    Google Scholar 

  • Trefethen L. N., The Asymptotic Accuracy of Rational Best Approximants to ez on a Disc, J. Approx. Theory 40 (1984), 380–383.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Walsh J. L., Interpolation and Approximation in the Complex Domain, Amer. Math. Soc. Colloq. Publns, 20 (1969), Amer. Math. Soc., Providence.

    Google Scholar 

References

  • Bauldry W. C., Máté A., Nevai P., Asymptotics for the solutions of systems of smooth recurrence equations, Pacific J. Math. 133 (1988), 209–227.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Freud G., On the coefficients of the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A (1) 76 (1976), 1–6.

    MathSciNet  MATH  Google Scholar 

  • Knopfmacher A., Lubinsky D. S., Nevai P., Freud’s conjecture and approximation of reciprocals of weights by polynomials, Constr. Approx. 4 (1988), 9–20.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lubinsky D. S., A survey of general orthogonal polynomials for weights on finite and infinite intervals, Acta Applicandæ Mathematicæ 10 (1987), 237–296.

    MathSciNet  MATH  Google Scholar 

  • Lubinsky D. S., Strong asymptotics for extremal errors and polynomials associated with Erdős-type weights, Pitman Res. Notes Math. 202 (1989), Longman.

    Google Scholar 

  • Lubinsky D. S., Mhaskar H. N., Saff E. B., A proof of Freud’s conjecture for exponential weights, Constr. Approx. 4 (1988), 65–83.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lubinsky D. S., Saff E. B. Strong Asymptotics for Extremal Polynomials Associated with Weights on ℝ, Lecture Notes Math., vol. 1305, Springer, 1988.

    Google Scholar 

  • Magnus, A. P., A proof of Freud’s conjecture about orthogonal polynomials related to |x|р exp(−x2m) for integer m, Lecture Notes Math., 1171, Springer 1985, pp. 362–372.

    CrossRef  MathSciNet  Google Scholar 

  • Magnus A. P., On Freud’s equations for exponential weights, J. Approx. Theory 46 (1986), 65–99.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Máté A., Nevai P., Zaslavsky T., Asymptotic expansion of ratios of coefficients of orthogonal polynomials with exponential weights, Trans. Amer. Math. Soc. 287 (1985), 495–505.

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Nevai P., Research problems in orthogonal polynomials, Approximation Theory VI, vol. 2, Academic Press, 1989, pp. 449–489.

    MathSciNet  MATH  Google Scholar 

  • Rahmanov E. A., On asymptotic properties of polynomials orthogonal on the real axis, Math. Sb. 119 (161) (1982), 163–203 (Russian); Math. USSR Sb. 47 (1984), 155–193. (English)

    Google Scholar 

  • Van Assche W., Asymptotics For Orthogonal Polynomials, Lect. Notes Math., vol. 1265, Springer, 1987.

    Google Scholar 

References

  • Gonchar A. A., Rahmanov E. A., Equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. 125 (1984), no. 167, 117–127. (Russian)

    MathSciNet  Google Scholar 

  • Ivanov K. G., Totik V., Fast decreasing polynomials, Constructive Approximation 6 (1990), 1–20

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Totik, V., Fast decreasing polynomials via potentials, manuscript.

    Google Scholar 

References

  • Freud G., On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), 1–6.

    MathSciNet  MATH  Google Scholar 

  • 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.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Van Assche W., Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987.

    CrossRef  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Van Assche W., Orthogonal polynomials on non-compact sets, Acad. Analecta, Med. Konink. Acad. Wetensch. Lett. Sch. Kunsten België, 51 (1989), no. 2, 1–36.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1994 Springer-Verlag

About this chapter

Cite this chapter

Nevai, P. (1994). Orthogonal polynomials. In: Havin, V.P., Nikolski, N.K. (eds) Linear and Complex Analysis Problem Book 3. Lecture Notes in Mathematics, vol 1574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101063

Download citation

  • DOI: https://doi.org/10.1007/BFb0101063

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57871-0

  • Online ISBN: 978-3-540-48368-7

  • eBook Packages: Springer Book Archive