Advertisement

Ukrainian Mathematical Journal

, Volume 46, Issue 8, pp 1070–1078 | Cite as

Some properties of biorthogonal polynomials and their application to Padé approximations

  • A. P. Golub
Article

Abstract

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N−1/N],N ε ℕ, for the functions
$$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$
wheref(z) is a function with known Padé approximants of the indicated orders,Tj[f;z] are Taylor polynomials of degreej for the functionf(z), and α m, M =\(\overline {1,M} \) are constants.

Keywords

Taylor Polynomial Biorthogonal Polynomial Biorthogonalizable Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. K. Dzyadyk, “On the generalization of the moment problem,”Dokl. Akad. Nauk Ukr.SSR, Ser. A, No. 6, 8–12 (1981).Google Scholar
  2. 2.
    A. P. Golub, “Some properties of biorthogonal polynomials,”Ukr. Mat. Zh.,41, No. 10, 1384–1388 (1989).Google Scholar
  3. 3.
    A. Iserles and S. P. Nørsett, “Biorthogonal polynomials,”Lect. Notes Math.,1171, 92–100 (1985).Google Scholar
  4. 4.
    A. Iserles and S. P. Nørsett, “On the theory of biorthogonal polynomials,”Math. Comput., No. 1, 42 (1986).Google Scholar
  5. 5.
    A. P. Golub,Application of the Generalized Moment Problem to the Padé Approximation of Some Functions [in Russian], Preprint No. 81.58, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1981), pp. 16–56.Google Scholar
  6. 6.
    P. K. Suétin,Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1979).Google Scholar
  7. 7.
    A. P. Golub, “On generalized moment representations of special type,”Ukr. Mat. Zh.,41, No. 11, 1455–1460 (1989).Google Scholar
  8. 8.
    R. Walliser, “Rationale Approximation desq-Analogous der Exponentialfunktion und Irrationalitätsaussagen für diese Funktion,”Arch. Math.,44, No. 1, 59–64 (1985).Google Scholar
  9. 9.
    H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol. 1, McGraw Hill, New York, Toronto, London (1953).Google Scholar
  10. 10.
    F. H. Jackson, “Transformation ofq-series,”Mess. Math.,39, 145–153 (1910).Google Scholar
  11. 11.
    E. Andrews and R. Askey, “Classical orthogonal polynomials,”Lect. Notes Math.,1171, 36–62 (1985).Google Scholar
  12. 12.
    A. P. Golub,Generalized Moment Representations and Rational Approximants [in Russian], Preprint No. 87.25, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).Google Scholar
  13. 13.
    G. A. Baker (Jr.) and P. Graves-Morris,Padé Approximants, Addison-Wesley, London, Amsterdam (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • A. P. Golub
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

Personalised recommendations