Skip to main content
SpringerLink
Log in

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

  • Published:
Constructive Approximation Aims and scope

Abstract

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aptekarev, A.I., Buslaev, V.I., Martínez-Finkelshtein, A., Suetin, S.P.: Padé approximants, continued fractions, and orthogonal polynomials. Russ. Math. Surv. 66(6), 37–122 (2011)

    Google Scholar 

  2. Aptekarev, A.I., López Lagomasino, G., Saff, E.B., Stahl, H., Totik, V.: Mathematical life of A.A. Gonchar (on his 80th birthday). Russ. Math. Surv. 66(6), 197–204 (2011)

    Google Scholar 

  3. Bolibrukh, A.A., Vitushkin, A.G., Vladimirov, V.S., Mishchenko, E.F., Novikov, S.P., Osipov, Yu.S., Sergeev, A.G., Ul’yanov, P.L., Faddeev, L.D., Chirka, E.M.: Andrei Aleksandrovich Gonchar (on his 70th birthday). Russ. Math. Surv. 57(1), 191–198 (2001)

    Article  Google Scholar 

  4. Cacoq, J., de la Calle Ysern, B., López Lagomasino, G.: Incomplete Padé approximation and convergence of row sequences of Hermite–Padé approximants. J. Approx. Theory (2012). doi:10.1016/j.jat.2012.05.005

    Google Scholar 

  5. Fidalgo Prieto, U., López Lagomasino, G.: Nikishin systems are perfect. Constr. Approx. 34, 297–356 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gonchar, A.A.: On convergence of Padé approximants for some classes of meromorphic functions. Sb. Math. 26, 555–575 (1975)

    Article  Google Scholar 

  7. Gonchar, A.A.: Poles of rows of the Padé table and meromorphic continuation of functions. Sb. Math. 43, 527–546 (1982)

    Article  MATH  Google Scholar 

  8. Gonchar, A.A.: Rational approximation of analytic functions. Proc. Steklov Inst. Math. 272(suppl. 2), S44–S57 (2011)

    Article  MathSciNet  Google Scholar 

  9. Hadamard, J.: Essai sur l’étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. 8, 101–186 (1892)

    Google Scholar 

  10. Hermite, Ch.: Sur la fonction exponentielle. C. R. Math. Acad. Sci. Paris 77, 18–24 (1873)

    MATH  Google Scholar 

  11. Hermite, Ch.: Sur la fonction exponentielle. C. R. Math. Acad. Sci. Paris 77, 74–79 (1873)

    Google Scholar 

  12. Hermite, Ch.: Sur la fonction exponentielle. C. R. Math. Acad. Sci. Paris 77, 226–233 (1873)

    Google Scholar 

  13. Hermite, Ch.: Sur la fonction exponentielle. C. R. Math. Acad. Sci. Paris 77, 285–293 (1873)

    Google Scholar 

  14. de Montessus de Ballore, R.: Sur les fractions continues algébriques. Bull. Soc. Math. Fr. 30, 28–36 (1902)

    MATH  Google Scholar 

  15. Graves-Morris, P.R., Saff, E.B.: A de Montessus theorem for vector-valued rational interpolants. In: Lecture Notes in Math., vol. 1105, pp. 227–242. Springer, Berlin (1984)

    Google Scholar 

  16. Graves-Morris, P.R., Saff, E.B.: Row convergence theorems for generalized inverse vector-valued Padé approximants. J. Comput. Appl. Math. 23, 63–85 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Graves-Morris, P.R., Saff, E.B.: An extension of a row convergence theorem for vector Padé approximants. J. Comput. Appl. Math. 34, 315–324 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nikishin, E.M., Sorokin, V.N.: Rational Approximations and Orthogonality. Transl. Math. Monogr., vol. 92. AMS, Providence (1991)

    MATH  Google Scholar 

  19. Sidi, A.: A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure. J. Approx. Theory 155, 75–96 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Suetin, S.P.: On poles of the mth row of a Padé table. Sb. Math. 48, 493–497 (1984)

    Article  MATH  Google Scholar 

  21. Suetin, S.P.: On an inverse problem for the mth row of the Padé table. Sb. Math. 52, 231–244 (1985)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The work of B. de la Calle Ysern received support from MINCINN under grant MTM2009-14668-C02-02 and from UPM through Research Group “Constructive Approximation Theory and Applications”. The work of J. Cacoq and G. López was supported by Ministerio de Economía y Competitividad under grants MTM2009-12740-C03-01 and MTM2012-36372-C03-01.

Author information

Authors and Affiliations

  1. Dpto. de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Universidad 30, 28911, Leganés, Spain

    J. Cacoq & G. López Lagomasino

  2. Dpto. de Matemática Aplicada, E. T. S. de Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006, Madrid, Spain

    B. de la Calle Ysern

Authors
  1. J. Cacoq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. de la Calle Ysern
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. López Lagomasino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. López Lagomasino.

Additional information

Communicated by Edward B. Saff.

In memory of A.A. Gonchar. He passed away on October 10, 2012 at the age of 80. See [2] and [3] for a brief account on his fruitful life and important contributions in approximation theory.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cacoq, J., de la Calle Ysern, B. & López Lagomasino, G. Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants. Constr Approx 38, 133–160 (2013). https://doi.org/10.1007/s00365-013-9188-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-013-9188-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation