Advertisement

From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals

  • Juan C. García-Ardila
  • Francisco MarcellánEmail author
  • Misael E. Marriaga
Conference paper
  • 46 Downloads
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

In this contribution, we present an overview of standard orthogonal polynomials by using an algebraic approach. Discrete Darboux transformations of Jacobi matrices are studied. Next, we emphasize the role of semiclassical orthogonal polynomials as a basic background to analyze sequences of polynomials orthogonal with respect to a Sobolev inner product. Some illustrative examples are discussed. Finally, we summarize some results in multivariate Sobolev orthogonal polynomials.

Keywords

Orthogonal polynomials Discrete Darboux transformations Semi-classical functionals Sobolev orthogonal polynomials 

Mathematics Subject Classification (2000)

Primary 42C05; Secondary 33C45 33D50 

Notes

Acknowledgements

The authors acknowledge the careful revision of the manuscript by the referees. Their comments and suggestions have contributed to improve its presentation.

The work of J. C. García-Ardila, F. Marcellán and M. Marriaga has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía, Industria y Competitividad of España, research project [MTM2015-65888-C4-2-P].

References

  1. 1.
    M. Alfaro, F. Marcellán, M.I. Rezola, A. Ronveaux, Sobolev type orthogonal polynomials: the nondiagonal case. J. Approx. Theory 83, 737–757 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Alfaro, J.J. Moreno-Balcázar, A. Peña, M.L. Rezola, A new approach to the asymptotics of Sobolev type orthogonal polynomials. J. Approx. Theory 163, 460–480 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    P. Althammer, Eine Erweiterung des orthogonalitätsbergriffes bei polynomen und dere anwendung auf die beste approximation. J. Reine Angew. Math. 211, 192–2014 (1962)MathSciNetzbMATHGoogle Scholar
  4. 4.
    R. Álvarez-Nodarse, J.J. Moreno-Balcázar, Asymptotic properties of generalized Laguerre orthogonal polynomials. Indag. Math. (N.S.) 15, 151–165 (2004)Google Scholar
  5. 5.
    J. Arvesú, M.J. Atia, F. Marcellán, On semiclassical linear functionals: the symmetric companion. Comm. Anal. Theory Contin. Fractions 10, 13–29 (2002)MathSciNetGoogle Scholar
  6. 6.
    K. Atkinson, O. Hansen, Solving the nonlinear Poisson equation on the unit disk. J. Integral Equations Appl. 17, 223–241 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    H. Bavinck, H.G. Meijer, Orthogonal polynomials with respect to a symmetric inner product involving derivatives. Appl. Anal. 33, 1003-117 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    H. Bavinck, H.G. Meijer, On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations. Indag. Math. (N.S.) 1, 7–14 (1990)Google Scholar
  9. 9.
    S. Belmehdi, Formes Linéaires et Polynômes Orthogonaux Semi-Classiques de Classs = 1. Description et Classification (Thèse d’État, Université Pierre et Marie Curie, Paris, 1990)Google Scholar
  10. 10.
    S. Belmehdi, On semiclassical linear functionals of class s = 1. Classification and integral representations. Indag. Math. (N.S.) 3(3), 253–275 (1992)Google Scholar
  11. 11.
    S. Bochner, Über Sturm-Liouvillesche polynomsysteme. Math. Z. 29(1), 730–736 (1929)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    L. Boelen, W. Van Assche, Discrete Painlevé equations for recurrence coefficients of semiclassical Laguerre polynomials. Proc. Amer. Math. Soc. 138, 1317–1331 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    J. Brenner, Über eine erweiterung des orthogonaltäts bei plynomen, in ed. by G. Alex its, S.B. Stechkin, Constructive Theory of Functions (Akdémiai Kiadó, Budapest, 1972), pp. 77–83Google Scholar
  14. 14.
    M.I. Bueno, F. Marcellán, Darboux transformation and perturbation of linear functionals. Linear Algebra Appl. 384, 215–242 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    A. Cachafeiro, F. Marcellán, J.J. Moreno-Balcázar, On asymptotic properties of Freud Sobolev orthogonal polynomials. J. Approx. Theory 25, 26–41 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    T.S. Chihara, On co-recursive orthogonal polynomials. Proc. Amer. Math. Soc. 8, 899–905 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    T.S. Chihara, An Introduction to Orthogonal Polynomials. Mathematics and its Applications Series, vol. 13 (Gordon and Breach Science Publishers, New York, 1978)Google Scholar
  18. 18.
    E.B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math. 55, 61–82 (1858)MathSciNetGoogle Scholar
  19. 19.
    P.A. Clarkson, K. Jordaan, Properties of generalized Freud polynomials. J. Approx. Theory 225, 148–175 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    P.A. Clarkson, K. Jordaan, A. Kelil, A generalized Freud-weight. Studies Appl. Math 136, 288–320 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    E.X.L. de Andrade, C.F. Bracciali, A. Sri Ranga, Zeros of Gegenbauer-Sobolev orthogonal polynomials: beyond coherent pairs. Acta Appl. Math. 105, 65–82 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    A.M. Delgado, F. Marcellán, Companion linear functionals and Sobolev inner procuts: a case study. Methods Appl. Anal. 11, 237–266 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    M. Derevyagin, J.C. Garcia-Ardila, F. Marcellán, Multiple Geronimus transformations. Linear Algebra Appl. 454, 158–183 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    H. Dueñas Ruiz, F. Marcellán, A. Molano, Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1) -coherent pairs. J. Math. Anal. Appl. 467(1), 601–621 (2018)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    C.F. Dunkl, Y. Xu, Orthogonal Polynomials in Several Variables. Encyclopedia of Mathematics and its Applications, vol. 155, 2nd edn. (Cambridge University Press, Cambridge, 2014)Google Scholar
  26. 26.
    A.J. Durán, A generalization of Favard’s theorem for polynomials satisfying a recurrence ration. J. Approx. Theory 74, 83–109 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    A.J. Durán, W. Van Assche, Orthogonal matrix polynomials and higher order recurrence relations. Linear Algebra Appl. 219, 261–280 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    W.D. Evans, L.L. Littlejohn, F. Marcellán, C. Markett, A. Ronveaux, On recrrence relations for Sobolev orthogonal polynomials. SIAM J. Math. Anal. 26, 446–467 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    L. Fernández, F. Marcellán, T.E. Pérez, M.A. Piñar, Y. Xu, Sobolev orthogonal polynomials on product domain. J. Comput. Appl. Math. 284, 202–215 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    J.S. Geronimo, D.S. Lubinsky, F. Marcellán, Asymptotic for Sobolev orthogonal polynomials for exponential weights. Constr. Approx. 22, 309–346 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Ya. L. Geronimus, On polynomials orthogonal with regard to a given sequence of numbers. Comm. Inst. Sci. Math. Mec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] 17(4), 3–18 (1940) (Russian)Google Scholar
  32. 32.
    Ya. L. Geronimus, On polynomials orthogonal with respect to numerical sequences and on Hahn’s theorem. Izv. Akad. Nauk. 4, 215–228 (1940) (in Russian)Google Scholar
  33. 33.
    W. Gröbner, in Orthogonale polynommsysteme, die gleichzeitig mit f(x) auch deren ableitung f′(x) approximieren, ed. by L. Collatz, Funktionalanalysis, Approximationstheorie, Numerische Mathematik, ISNM, vol. 7 (Birkhäuser, Basel, 1967), pp. 24–32Google Scholar
  34. 34.
    W. Hahn, Über Differentialgleichunqen für Orthoqonalpolynome. Monat. Math. 95, 269–274 (1983)CrossRefGoogle Scholar
  35. 35.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  36. 36.
    A. Iserles, J.M. Sanz-Serna, P.E. Koch, S.P. Nørsett, Orthogonality and Approximation in a Sobolev Space, in Algorithms for Approximation, II (Shrivenham, 1988) (Chapman and Hall, London, 1990), pp. 117–124zbMATHGoogle Scholar
  37. 37.
    M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)Google Scholar
  38. 38.
    D.H. Kim, K.H. Kwon, F. Marcellán, G.J. Yoon, Sobolev orthogonality and coherent pairs of moment functionals: an inverse problem. Internat. Math. J. 2, 877–888 (2002)MathSciNetzbMATHGoogle Scholar
  39. 39.
    R. Koekoek, Koornwinder’s Laguerre polynomials. Delft Prog. Rep. 12, 393–404 (1988)MathSciNetzbMATHGoogle Scholar
  40. 40.
    R. Koekoek, Generalizations of Laguerre polynomials. J. Math. Anal. Appl. 153, 576–590 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    R. Koekoek, H.G. Meijer, A generalization of Laguerre polynomials. SIAM J. Math. Anal. 24, 768–782 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    R. Koekoek, P. Lesky, R. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics (Springer, Berlin, 2010)Google Scholar
  43. 43.
    H.L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation. Pa. State Coll. Stud. 6, 1–24 (1940)zbMATHGoogle Scholar
  44. 44.
    A.M. Krall, Orthogonal polynomials satisfying fourth order differential equations. Proc. Roy. Soc. Edinburgh, Sect. A 87, 271–288 (1980/1981)Google Scholar
  45. 45.
    D.C. Lewis, Polynomial least square approximations. Amer. J. Math. 69, 273–278 (1947)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    H. Li, Y. Xu, Spectral approximation on the unit ball. SIAM J. Numer. Anal. 52(6), 2647–2675 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    G. López Lagomasino, H. Pijeira, Nth root asymptotics of Sobolev orthogonal polynomials. J. Approx. Theory 99, 30–43 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    G. López Lagomasino, F. Marcellán, W. Van Assche, Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product. Constr. Approx. 11, 107–137 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  49. 49.
    F. Marcellán, P. Maroni, Sur l’adjonction d’une masse de Dirac à une forme régulière et semi-classique. Ann. Mat. Pura Appl. 162(4), 1–22 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    F. Marcellán, J.J. Moreno-Balcázar, Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports. Acta Appl. Math. 94, 163–192 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    F. Marcellán, T.E. Pérez, M.A. Piñar, Laguerre Sobolev orthogonal polynomials. J. Comput. Appl. Math. 71, 245–265 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  52. 52.
    F. Marcellán, J. Petronilho, Orthogonal polynomials and coherent pairs: the classical case. Indag. Math. (N.S.) 6, 287–307 (1995)Google Scholar
  53. 53.
    F. Marcellán, A. Ronveaux, On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math. (N.S.) 1, 451–464 (1990)Google Scholar
  54. 54.
    F. Marcellán, W. Van Assche, Relative asymptotics for orthogonal polynomials with a Sobolev inner product. J. Approx. Theory 72, 193–209 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    F. Marcellán, R. Xh. Zejnullahu, B. Xh. Fejzullahu, E. Huertas, On orthogonal polynomials with respect to certain discrete Sobolev inner product. Pacific J. Math. 257, 167–188 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    F. Marcellán, Y. Xu, On Sobolev orthogonal polynomials. Expo. Math. 33(3), 308–352 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  57. 57.
    P. Maroni, Une caractérisation des polynômes orthogonaux semi-classiques, C. R. Acad. Sci. Paris Sér. I Math. 301(6), 269–272 (1985)MathSciNetzbMATHGoogle Scholar
  58. 58.
    P. Maroni, Le calcul des formes linéaires et les polynômes orthogonaux semi-classiques, in ed. by M. Alfaro, Orthogonal Polynomials and Their Applications, LNM 1329 (Springer, Berlin, 1988), pp. 279–290Google Scholar
  59. 59.
    P. Maroni, Une thèorie algébrique des polynômes orthogonaux. Application aux polynômes semiclassiques, in Orthogonal Polynomials and Their Applications (Erice 1990). IMACS Annals of Computational and Applied Mathematics, vol. 9 (Baltzer, Basel, 1991), pp. 95–130Google Scholar
  60. 60.
    A. Martínez-Finkelshtein, Bernstein-Szegő’s theorem for Sobolev orthogonal polynomials. Constr. Approx. 16, 73–84 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  61. 61.
    A. Martínez-Finkelshtein, J.J. Moreno-Balcázar, T.E. Pérez, M.A. Piñar, Asymptotics of Sobolev orthogonal polynomials for coherent pairs. J. Approx. Theory 92, 280–2893 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  62. 62.
    H.G. Meijer, A short history of orthogonal polynomials in a Sobolev space I. The non-discrete case, in: 31st Dutch mathematical conference, Groningen, 1995. Nieuw Arch. Wiskd. 14(4), 93–112 (1996)Google Scholar
  63. 63.
    H.G. Meijer, Determination of all coherent pairs of functionals. J. Approx. Theory 89, 321–343 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics (Birkhäuser Verlag, Basel 1988)CrossRefzbMATHGoogle Scholar
  65. 65.
    T.E. Pérez, M.A. Piñar, Y. Xu, Weighted Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 171, 84–104 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  66. 66.
    M.A. Piñar, Y. Xu, Orthogonal polynomials and partial differential equations on the unit ball. Proc. Amer. Math. Soc. 37, 2979–2987 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  67. 67.
    F.W. Schäfke, Zu den orthogonalplynomen von Althammer. J. Reine Angew. Math. 252, 195–199 (1972)MathSciNetzbMATHGoogle Scholar
  68. 68.
    F.W. Schäfke, G. Wolf, Einfache verallgemeinerte klassische orthogonal polynome. J. Reine Angew. Math. 262/263, 339–355 (1973)Google Scholar
  69. 69.
    J. Shohat, A differential equation for orthogonal polynomials. Duke Math. J. 5, 401–417 (1939)CrossRefMathSciNetzbMATHGoogle Scholar
  70. 70.
    G. Szegő, Orthogonal Polynomials, 4th edn., vol. 23. American Mathematical Society, Colloquium Publications (American Mathematical Society, Providence, 1975)Google Scholar
  71. 71.
    V.B. Uvarov, Relation between polynomials orthogonal with different weights. Doklady Akademii Nauk SSSR 126(1), 33–36 (1959) (in Russian)MathSciNetzbMATHGoogle Scholar
  72. 72.
    V.B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions. USSR Comput. Math. Phys. 9, 25–36 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  73. 73.
    W. Van Assche, Orthogonal Polynomials and Painlevé Equations. Australian Mathematical Society Lecture Series, vol. 27 (Cambridge University Press, Cambridge, 2018)Google Scholar
  74. 74.
    Y. Xu, A family of Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 138, 232–241 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  75. 75.
    Y. Xu, Sobolev orthogonal polynomials defined via gradient on the unit ball. J. Approx. Theory 152, 52–65 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  76. 76.
    G. Yoon, Darboux transforms and orthogonal polynomials. Bull. Korean Math. Soc. 39, 359–376 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  77. 77.
    A. Zhedanov, Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85, 67–86 (1997)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Juan C. García-Ardila
    • 1
  • Francisco Marcellán
    • 2
    • 3
    Email author
  • Misael E. Marriaga
    • 4
  1. 1.Departamento de Matemática Aplicada a la Ingeniería IndustrialUniversidad Politécnica de MadridMadridSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain
  3. 3.Instituto de Ciencias Matemáticas (ICMAT)MadridSpain
  4. 4.Departamento de Matemática Aplicada, Ciencia e Ingeniería de Materiales y Tecnología ElectrónicaUniversidad Rey Juan CarlosMóstolesSpain

Personalised recommendations