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A higher order Sobolev-type inner product for orthogonal polynomials in several variables

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Abstract

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived.

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References

  1. Abramowitz, M., Stegun, I. A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Ninth printing. Dover New York (1972).

  2. Alfaro, M., Marcellán, F., Rezola, M.L., Ronveaux, A.: On orthogonal polynomials of Sobolev type: algebraic properties and zeros. SIAM J. Math. Anal. 23(3), 737–757 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Álvarez-Nodarse, R., Moreno-Balcázar, J.J.: Asymptotic properties of generalized Laguerre orthogonal polynomials. Indag. Math. N. S. 15, 151–165 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barlett, M.S.: An inverse matrix adjustment arising in discriminant analysis. Ann. Math. Statist. 22(1), 107–111 (1951)

    Article  Google Scholar 

  5. Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of several variables. Encyclopedia of Mathematics and its Applications, vol. 81. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  6. Marcellán, F., Ronveaux, A.: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math. N. S. 1, 451–464 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Marcellán, F., Van Assche, W.: Relative asymptotics for orthogonal polynomials with a Sobolev inner product. J. Approx. Theory 72, 193–209 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Martínez-Finkelshtein, A.: Asymptotic properties of Sobolev orthogonal polynomials. J. Comput. Appl. Math. 99, 491–510 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Martínez-Finkelshtein, A.: Asymptotic properties of Sobolev orthogonal polynomials revisited. J. Comput. Appl. Math. 127, 255–266 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mello, M.V., Paschoa, V.C., Pérez, T.E., Piñar, M.A.: Multivariate Sobolev-type orthogonal polynomials. Jaen J. Approx. 3(2), 241–259 (2011)

    MATH  MathSciNet  Google Scholar 

  11. Szegő, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ. vol. 23 (1975). Amer. Math. Soc., Providence Rhode Island (1975)

  12. Xu, Y.: Asymptotics for orthogonal polynomials and Christoffel functions on a ball. Methods Appl. Anal. 3(2), 257–272 (1996)

    MATH  MathSciNet  Google Scholar 

  13. Xu, Y.: Summability of Fourier orthogonal series for Jacobi weights on a ball in ℝd. Trans. Amer. Math. Soc. 351(6), 2439–2458 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Nacional de Colombia, Cd. Universitaria, Bogotá, Colombia

    Herbert Dueñas

  2. Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, México

    Luis E. Garza

  3. Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Miguel Piñar

Authors
  1. Herbert Dueñas
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  2. Luis E. Garza
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  3. Miguel Piñar
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Correspondence to Luis E. Garza.

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Dueñas, H., Garza, L.E. & Piñar, M. A higher order Sobolev-type inner product for orthogonal polynomials in several variables. Numer Algor 68, 35–46 (2015). https://doi.org/10.1007/s11075-014-9836-x

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  • DOI: https://doi.org/10.1007/s11075-014-9836-x

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