Computational Methods and Function Theory

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

Convergence of Row Sequences of Simultaneous Padé-orthogonal Approximants



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).


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 



I wish to express my gratitude toward the anonymous referee and the editor for helpful comments and suggestions leading to improvements of this work.


  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

Personalised recommendations