Computational Methods and Function Theory

, Volume 17, Issue 3, pp 525–556 | Cite as

Convergence of Row Sequences of Simultaneous Padé-orthogonal Approximants

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Abstract

In this paper, convergence theorems of row sequences of vector valued Padé-orthogonal approximants (simultaneous Padé-orthogonal approximants) corresponding to a measure supported on a general compact subset of the complex plane are proved. These theorems are natural extensions of Montessus de Ballore’s theorem for row sequences of (scalar) Padé-orthogonal approximants in Bosuwan et al. (Jaen J Approx 5:179–208, 2013).

Keywords

Montessus de Ballore’s theorem Frobenius–Padé approximants of orthogonal expansions Padé-orthogonal approximants Fourier–Padé approximants Simultaneous Padé approximants Orthogonal polynomials Hermite-Padé approximants 

Mathematics Subject Classification

30E10 41A21 

References

  1. 1.
    Barrios Rolanía, D., López Lagomasino, G.: Ratio asymptotics for polynomials orthogonal on arcs of the unit circle. Constr. Approx. 15, 1–31 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bello Hernández, M., López Lagomasino, G.: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle. J. Approx. Theory 92, 216–244 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bosuwan, N., López Lagomasino, G., Saff, E.B.: Determining singularities using row sequences of Padé-orthogonal approximants. Jaen J. Approx. 5, 179–208 (2013)MathSciNetMATHGoogle Scholar
  4. 4.
    Bosuwan, N., López Lagomasino, G.: Theorem on row sequences of linear Padé-orthogonal approximants. Comput. Methods Funct. Theory 15(4), 529–554 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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 170, 59–77 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cacoq, J., López Lagomasino, G.: Convergence of row sequences of simultaneous Fourier–Padé approximation. Jaen J. Approx. 4, 101–120 (2012)MathSciNetMATHGoogle Scholar
  7. 7.
    Carleman, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Ark. Mat. Astron. Fys. 17, 215–244 (1923)MATHGoogle Scholar
  8. 8.
    Gonchar, A.A., Grigorjan, L.D.: On estimates of the norm of the holomorphic component of a meromorphic function. Sb. Math. 28, 571–575 (1976)CrossRefGoogle Scholar
  9. 9.
    Gonchar, A.A.: On the convergence of generalized Padé approximants of meromorphic functions. Math. USSR Sb. 140, 564–577 (1975)Google Scholar
  10. 10.
    Gonchar, A.A.: Poles of rows of the Padé table and meromorphic continuation of functions. Sb. Math. 43, 527–546 (1981)CrossRefMATHGoogle Scholar
  11. 11.
    Graves-Morris, P.R., Saff, E.B.: A de Montessus theorem for vector-valued rational interpolants. Lecture Notes in Math, vol. 1105. Springer, Berlin (1984)Google Scholar
  12. 12.
    Kaliaguine, V.A.: A note on the asymptotics of orthogonal polynomials on a complex arc: the case of a measure with a discrete part. J. Approx. Theory 80, 138–145 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Miña-Díaz, E.: An asymptotic integral representation for Carleman orthogonal polynomials. Int. Math. Res. Not. IMRN 2008 (2008) (Art. ID rnn065)Google Scholar
  14. 14.
    Miña-Díaz, E.: An expansion for polynomials orthogonal over an analytic Jordan curve. Commun. Math. Phys. 285, 1109–1128 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Miña-Díaz, E.: Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight. J. Math. Anal. Appl. 372, 306–315 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, vol. 54, Parts I and II. Colloquium Publications, American Mathematical Society, Providence (2005)Google Scholar
  17. 17.
    Simon, B.: Szegő’s Theorem and Its Descendants. Princeton University Press, Princeton (2011)MATHGoogle Scholar
  18. 18.
    Sobczyk, G.: Generalized Vandermonde determinants and applications. Aportaciones Matematicas, Serie Comunicaciones 30, 203–213 (2002)MathSciNetMATHGoogle Scholar
  19. 19.
    Stahl, H., Totik, V.: General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge (1992)Google Scholar
  20. 20.
    Stylianopoulos, N.: Strong asymptotics for Bergman polynomials over domains with corners and applications. Constr. Approx. 38, 59–100 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Suetin, P.K.: Fundamental properties of polynomials orthogonal on a contour. Russ. Math. Surv. 21, 35–83 (1966)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Suetin, P.K.: Polynomials Orthogonal over a Region and Bieberbach Polynomials. American Mathematical Society, Providence (1974)MATHGoogle Scholar
  23. 23.
    Suetin, S.P.: On the convergence of rational approximations to polynomial expansions in domains of meromorphy of a given function. Math. USSR Sb. 34, 367–381 (1978)CrossRefMATHGoogle Scholar
  24. 24.
    Szegő, G.: Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören. Math. Z. 9, 218–270 (1921)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Szegő, G.: Orthogonal Polynomials, vol. 23, 4th edn. Amer. Math. Soc. Colloq. Publ. American Mathematical Society, Providence (1975)Google Scholar
  26. 26.
    Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMahidol UniversityBangkokThailand
  2. 2.Centre of Excellence in MathematicsCHEBangkokThailand

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