Advertisement

Computational Methods and Function Theory

, Volume 15, Issue 4, pp 529–554 | Cite as

Inverse Theorem on Row Sequences of Linear Padé-orthogonal Approximation

  • N. BosuwanEmail author
  • G. López Lagomasino
Article

Abstract

We give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Padé-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we obtain an analog of Gonchar’s theorem on row sequences of Padé approximants.

Keywords

Padé approximation Padé-orthogonal approximation  Orthogonal polynomials Fourier–Padé approximation Inverse problems 

Mathematics Subject Classification

Primary 30E10 41A27 Secondary 41A21 

References

  1. 1.
    Baker, G.A., Graves-Morris, P.: Padé approximants, 2nd edn. In: Encyclopedia of Mathematics and its Applications, vol. 59. Cambridge University, Cambridge (1996)Google Scholar
  2. 2.
    Rolanía, D.B., López Lagomasino, G.: Ratio asymptotics for polynomials orthogonal on arcs of the unit circle. Constr. Approx. 15, 1–31 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Rolanía, D.B., López Lagomasino, G., Saff, E.B.: Asymptotics of orthogonal polynomials inside the unit circle and Szegő–Padé approximants. J. Comput. Appl. Math. 133, 171–181 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Rolanía, D.B., López Lagomasino, G., Saff, E.B.: Determining radii of meromorphy via orthogonal polynomials on the unit circle. J. Approx. Theory 124, 263–281 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hernández, M.B., López Lagomasino, G.: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle. J. Approx. Theory 92, 216–244 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bosuwan, N., López Lagomasino, G., Saff, E.B.: Determining singularities using row sequences of Padé-orthogonal approximants. Jaen J. Approx. 5, 179–208 (2013)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Buslaev, V.I.: On the Fabry ratio theorem for orthogonal series. Complex analysis and applications. Proc. Steklov Inst. Math. 253, 8–21 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buslaev, V.I.: An analogue of Fabry’s theorem for generalized Padé approximants. Math. Sb. 200, 39–106 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cacoq, J., López Lagomasino., G.: Convergence of row sequences of simultaneous Fourier-Padé approximation. Jaen J. Approx. 4, 101–120 (2012)zbMATHMathSciNetGoogle Scholar
  10. 10.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Carleman, T.: Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen. Ark. Mat. Astron. Fys. 17, 215–244 (1923)Google Scholar
  12. 12.
    Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  13. 13.
    Fleischer, J.: Generalizations of Padé Approximants. Los Alamos Scientific Laboratory, New Mexico (1972)Google Scholar
  14. 14.
    Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. Transl. of Math. Monographs, vol. 36. American Mathematical Society, Providence (1969)Google Scholar
  15. 15.
    Gonchar, A.A.: On convergence of Padé approximants for some classes of meromorphic functions. Math. USSR Sb. 26, 555–575 (1975)CrossRefGoogle Scholar
  16. 16.
    Gonchar, A.A.: On the convergence of generalized Padé approximants of meromorphic functions. Math. USSR Sb. 27, 503–514 (1975)CrossRefGoogle Scholar
  17. 17.
    Gonchar, A.A., Rakhmanov, E.A., Suetin, S.P.: On the convergence of Padé approximations of orthogonal expansions. Proc. Steklov Inst. Math. 200, 149–159 (1993)MathSciNetGoogle Scholar
  18. 18.
    Gonchar, A.A., Rakhmanov, E.A., Suetin, S.P.: On the rate of convergence of Padé approximants of orthogonal expansions. In:Progress in Approximation Theory, pp. 169–190. Springer, New York (1992)Google Scholar
  19. 19.
    Gonchar, A.A.: Rational approximation of analytic functions. Proc. Steklov Inst. Math. 272, S44–S57 (2011)zbMATHCrossRefGoogle Scholar
  20. 20.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Lubinsky, D.S., Sidi, A.: Convergence of linear and nonlinear Padé approximants from series of orthogonal polynomials. Trans. Am. Math. Soc. 278, 333–345 (1983)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Maehly, H.J.: Rational approximations for transcendental functions. In: Proceedings of the International Conference on Information Processing, Butterworths, pp. 57–62 (1960)Google Scholar
  23. 23.
    Miña-Díaz, E.: An asymptotic integral representation for Carleman orthogonal polynomials. Int. Math. Res. Not. 2008, 1–38 (2008)Google Scholar
  24. 24.
    Miña-Díaz, E.: An expansion for polynomials orthogonal over an analytic Jordan curve. Commun. Math. Phys. 285, 1109–1128 (2009)zbMATHCrossRefGoogle Scholar
  25. 25.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Simanek, B.: Ratio asymptotics, Hessenberg matrices, and weak asymptotic measures. Int. Math. Res. Not. 24, 6798–6825 (2014)MathSciNetGoogle Scholar
  27. 27.
    Simon, B.: Orthogonal Polynomials on the Unit Circle, vol. 54, Parts I and II. Colloquium Publications, American Mathematical Society, Providence (2005)Google Scholar
  28. 28.
    Simon, B.: Szegő’s Theorem and its Descendants. Princeton University Press, Princeton (2011)Google Scholar
  29. 29.
    Sobczyk, G.: Generalized Vandermonde determinants and applications. Aportaciones Matematicas, Serie Comunicaciones 30, 203–213 (2002)MathSciNetGoogle Scholar
  30. 30.
    Stahl, H., Totik, V.: General orthogonal polynomials. In: Encyclopedia of Mathematics and its Applications, vol. 43. Cambridge University Press, Cambridge (1992)Google Scholar
  31. 31.
    Stylianopoulos, N.: Strong asymptotics for Bergman polynomials over domains with corners and applications. Constr. Approx. 38, 59–100 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Suetin, P.K.: Fundamental properties of polynomials orthogonal on a contour. Russ. Math. Surv. 21, 35–83 (1966)zbMATHCrossRefGoogle Scholar
  33. 33.
    Suetin, P.K.: Polynomials Orthogonal over a Region and Bieberbach Polynomials. American Mathematical Society, Providence (1974)Google Scholar
  34. 34.
    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)zbMATHCrossRefGoogle Scholar
  35. 35.
    Suetin, S.P.: Inverse theorems on generalized Padé approximants. Math. USSR Sb. 37, 581–597 (1980)zbMATHCrossRefGoogle Scholar
  36. 36.
    Suetin, S.P.: On Montessus de Ballore’s theorem for rational approximants of orthogonal expansions. Math. USSR Sb. 42, 399–411 (1982)zbMATHCrossRefGoogle Scholar
  37. 37.
    Suetin, S.P.: On an inverse problem for the \(m\)th row of the Padé table. Sb. Math. 52, 231–244 (1985)zbMATHCrossRefGoogle Scholar
  38. 38.
    Suetin, S.P.: Asymptotics of the denominators of the diagonal Padé approximations of orthogonal expansions. Dokl. Ross. Akad. Nauk. 56, 774–776 (1997)Google Scholar
  39. 39.
    Suetin, S.P.: Padé approximants and efficient analytic continuation of a power series. Russ. Math. Surv. 57, 43–141 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Szegő, G.: Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören. Math. Z. 9, 218–270 (1921)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Szegő, G.: Orthogonal Polynomials, vol. 23, 4th edn. Amer. Math. Soc. Colloq. Publ. American Mathematical Society, Providence (1975)Google Scholar
  42. 42.
    Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMahidol UniversityBangkokThailand
  2. 2.Centre of Excellence in Mathematics, CHEBangkokThailand
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

Personalised recommendations