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Shohat-Favard and Chebyshev’s methods in d-orthogonality

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Abstract

This paper is concerned with the Shohat-Favard, Chebyshev and Modified Chebyshev methods for d-orthogonal polynomial sequences d∈ℕ. Shohat-Favard’s method is presented from the concept of dual sequence of a sequence of polynomials. We deduce the recurrence relations for the Chebyshev and the Modified Chebyshev methods for every d∈ℕ. The three methods are implemented in the Mathematica programming language. We show several formal and numerical tests realized with the software developed.

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References

  1. C. Brezinski, Biorthogonality and its Applications to Numerical Analysis (Marcel Dekker, New York, 1991).

    Google Scholar 

  2. C. Brezinski and J. Van Iseghem, Vector orthogonal polynomials of dimension _d, in: Approximation and Computation, ed. R.V.M. Zahar, International Series of Numerical Mathematics 115 (1994) pp. 29–39.

  3. P.L. Chebyshev, Sur les fractions continues, J. Math. Pures Appl. (2) 3 (1858) 289–323.

    Google Scholar 

  4. K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (1996) 279–295.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Douak and P. Maroni, Les polynômes orthogonaux classiques de dimension deux, Analysis 12 (1992) 71–107.

    MATH  MathSciNet  Google Scholar 

  6. K. Douak and P. Maroni, On d-orthogonal Tchebychev polynomials I, Appl. Numer. Math. 24 (1997) 23–53.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Favard, C. R. Acad. Sci. Paris 200 (1935) 2053.

    MATH  Google Scholar 

  8. W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982) 289–317.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Gautschi, Orthogonal polynomials – constructive theory and applications, J. Comput. Appl. Math. 12/13 (1985) 61–76.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Maroni, L’orthogonalité et les récurrences de polynômes d’ordre supérieure à deux, Ann. Fac. Sci. Toulouse Math. 10(1) (1989) 105–139.

    MATH  MathSciNet  Google Scholar 

  11. P. Maroni, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 (1993) 133–155.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Maroni, Fonctions Eulériennes. Polynômes orthogonaux classiques, in: Techniques de l’Ing´enieur, Trait´e G´en´eralis´e, Sciences Fondamentales (1994).

  13. M. Morandi Cecchi and M. Redivo-Zaglia, Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments, J. Comput. Appl. Math. 49 (1993) 207–216.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Prudnikov, Orthogonal polynomials with ultra-exponential weight functions, J. Comput. Appl. Math. 48 (1993) 239–241.

    MathSciNet  Google Scholar 

  15. M. Redivo-Zaglia, Extrapolation, méthodes de Lanczos et polynômes orthogonaux: Théorie et conception de logiciels, Thèse, Université des Sciences et Technologies de Lille (1992).

  16. J.A. Shohat, C. R. Acad. Sci. Paris 207 (1938) 556.

    MATH  Google Scholar 

  17. J. Van Iseghem, Vector Padé approximants (IMACS Oslo 1985), in: Numer. Math. Appl., eds. R. Vichnevestsky and J. Vignes (Elsevier/North-Holland, Amsterdam, 1986) pp. 73–77.

    Google Scholar 

  18. J. Van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math. 19 (1987) 141–150.

    MATH  MathSciNet  Google Scholar 

  19. J. Van Iseghem, Approximants de Padé vectoriels, Thése d’Etat, Universit´e des Sciences et Techniques de Lille-Flandres Artois (1987).

  20. J.C. Wheeler, Modified moments and Gaussian quadrature, Rocky Mountain J. Math. 4 (1974) 287–296.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media / Cambridge Univ. Press, 1996).

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Authors
  1. Zélia da Rocha
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da Rocha, Z. Shohat-Favard and Chebyshev’s methods in d-orthogonality. Numerical Algorithms 20, 139–164 (1999). https://doi.org/10.1023/A:1019151817161

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