Orthogonal polynomials on the real line

  • Gradimir V. Milovanović
Part of the Contemporary Mathematicians book series (CM)


In about two dozen papers, Walter Gautschi developed the so-called constructive theory of orthogonal polynomials on ℝ, ncluding effective algorithms for numerically generating orthogonal polynomials, a detailed stability analysis of such algorithms as well as several new applications of orthogonal polynomials. Furthermore, he provided software necessary for implementing these algorithms (see Section 23, Vol. 3) and applications.


  1. [1]
    A. I. Aptekarev. Multiple orthogonal polynomials. Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997). J. Comput. Appl. Math., 99(1-2):423-447, 1998.Google Scholar
  2. [2]
    G. I. Barkov. Some systems of polynomials orthogonal in two symmetric intervals. Izv. Vysš. Učebn. Zaved. Matematika, 17(4):3-16, 1960. (Russian)Google Scholar
  3. [3]
    Carlos F. Borges. On a class of Gauss-like quadrature rules. Numer. Math., 67(3): 271-288, 1994.MATHGoogle Scholar
  4. [4]
    E. B. Christoffel. Über die Gaußische Quadratur und eine Verallgemeinerung derselben. J Reine Angew. Math., 55:61-82, 1858; Ges. Math. Abhandlungen I, 65-87.Google Scholar
  5. [5]
    Aleksandar S. Cvetković and Gradimir V. Milovanović. The Mathematica package “OrthogonalPolynomials". Facta Univ. Ser. Math. Inform., 19:17-36, 2004.Google Scholar
  6. [6]
    A. S. Cvetković, G. V. Milovanović, and M. M. Matejić. Rational algorithm for quadratic Christoffel modification and applications to the constrained L2-approximation. Int. J. Comput. Math., 88(14):3012-3025, 2011.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Germund Dahlquist. On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. I. BIT, 37(2):256-295, 1997.Google Scholar
  8. [8]
    Germund Dahlquist. On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. II. BIT, 37(4):804-832, 1997.Google Scholar
  9. [9]
    Germund Dahlquist. On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules. III. BIT, 39(1):51-78, 1999.Google Scholar
  10. [10]
    Arne Fransén. Accurate determination of the inverse gamma integral. BIT, 19(1): 137-138, 1979.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    David Galant. An implementation of Christoffel’s theorem in the theory of orthogonal polynomials. Math. Comp., 25(113):111-113, 1971.Google Scholar
  12. [12]
    Gene H. Golub and John H. Welsch. Calculation of Gauss quadrature rules. Math. Comp., 23(106):221-230, 1969; addendum, ibid, loose microfiche suppl. A1-A10.Google Scholar
  13. [13]
    Soo-Y. Lee. The inhomogeneous Airy functions, Gi(z) and Hi(z). J. Chem. Phys., 72(1):332-336, 1980.Google Scholar
  14. [14]
    F. Locher. A family of generalized Jacobi polynomials. Math. Comp., 53(187):303-309,1989.Google Scholar
  15. [15]
    F. Marcellán, M. Alfaro, and M. L. Rezola. Orthogonal polynomials on Sobolev spaces: old and new directions. Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991). J. Comput. Appl. Math., 48(1-2):113-131, 1993.Google Scholar
  16. [16]
    F. Marcellán and A. Ronveaux. On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math. (N.S.), 1(4):451-464, 1990.Google Scholar
  17. [17]
    Giuseppe Mastroianni and Gradimir V. Milovanović. Interpolation processes. Basic theory and applications. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008, xiv+444 pp. ISBN: 978-3-540-68346-9.Google Scholar
  18. [18]
    Gradimir V. Milovanović. Construction of s-orthogonal polynomials and Turán quadrature formulae. In Numerical methods and approximation theory, III (Niš, 1987),311-328, Univ. Niš, Niš, 1988.Google Scholar
  19. [19]
    Gradimir V. Milovanović. Summation of series and Gaussian quadratures. In Approximation and computation (West Lafayette, IN, 1993), 459-475, Internat. Ser. Numer. Math., 119, Birkhäuser Boston, Boston, MA, 1994.Google Scholar
  20. [20]
    Gradimir V. Milovanović. Summation of slowly convergent series via quadratures. In Proceedings of the Third International Conference on Advances in Numerical Methods and Applications O(h3), Sofia, Bulgaria, 21-26 August, 1994 (Dimov, I.T., ed. et al.), World Scientific, 154-161, Singapore, 1994.Google Scholar
  21. [21]
    Gradimir V. Milovanović. Summation of series and Gaussian quadratures.II. Special functions (Torino, 1993). Numer. Algorithms, 10(1-2):127-136, 1995.Google Scholar
  22. [22]
    Gradimir V. Milovanović. A class of orthogonal polynomials on the radial rays in the complex plane. J. Math. Anal. Appl., 206(1):121-139, 1997.MATHGoogle Scholar
  23. [23]
    Gradimir V. Milovanović. Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation. In Numerical Analysis 2000 , Vol.V, Quadra-ture and Orthogonal Polynomials. (W. Gautschi, F. Marcellán,and L. Reichel, eds.), J. Comput. Appl. Math., 127(1-2):267-286, 2001.Google Scholar
  24. [24]
    Gradimir V. Milovanović. Orthogonal polynomials on the radial rays in the complex plane and applications. In Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, Vol. I (Potenza, 2000). Rend. Circ. Mat. Palermo (2), Suppl., 68:65-94, 2002.Google Scholar
  25. [25]
    Gradimir V. Milovanović, Methods for computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures. Jaen J. Approx., to appear.Google Scholar
  26. [26]
    Gradimir V. Milovanović and Aleksandar S. Cvetković. Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. Math. Balkanica, 26(1-2):169-184, 2012.Google Scholar
  27. [27]
    Gradimir V. Milovanović and Marija Stanić. Construction of multiple orthogonal polynomials by discretized Stieltjes-Gautschi procedure and corresponding Gaussian quadratures. Facta Univ. Ser. Math. Inform., 18:9-29, 2003.MATHGoogle Scholar
  28. [28]
    R. Piessens and M. Branders. Tables of Gaussian quadrature formulas. Appl. Math. Progr. Div. University of Leuven, Leuven, 1975.Google Scholar
  29. [29]
    R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments. Numer. Math., 18(5):465-478, 1972.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    N. M. Steen, G. D. Byrne, and E. M. Gelbard. Gaussian quadratures for the integrals \(\int^{\infty}_{0}\text{exp}(-x^{2})f(x)dx \text{ and} \int^{b}_{0}\) exp(−x2)f(x) dx. Math. Comp., 23(107):661–671, 1969.Google Scholar
  31. [31]
    V.B. Uvarov. Relation between polynomials orthogonal with different weights. Dokl. Akad. Nauk SSSR, 126(1):33–36, 1959. (Russian).Google Scholar
  32. [32]
    Pierre Verlinden. Stable rational modification of a weight. Numer. Algorithms, 22(2): 183-192, 1999.Google Scholar
  33. [33]
    John C. Wheeler. Modified moments and Gaussian quadratures. In Proceedings of the International Conference of Padé Approximants, Continued Fractions and Related Topics (Univ. Colorado, Boulder, Colo., 1972; dedicated to the memory of H.S. Wall). Rocky Mountain J. Math., 4(2):287-296, 1974.Google Scholar
  34. [34]
    John C. Wheeler. Modified moments and continued fraction coefficients for the diatomic linear chain. J. Chem. Phys., 80(1):472-476, 1984.Google Scholar
  35. [35]
    R. Wong. Quadrature formulas for oscillatory integral transforms. Numer. Math., 39(3):351-360, 1982.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Gradimir V. Milovanović
    • 1
  1. 1.Matematički Institut SANUBeogradSerbia

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