Advertisement

Siberian Mathematical Journal

, Volume 56, Issue 3, pp 442–454 | Cite as

Convergence conditions for interpolation rational fractions with finitely many poles

  • A. G. LipchinskiĭEmail author
Article
  • 20 Downloads

Abstract

We consider an interpolation process for the class of functions with finitely many singular points by means of rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes form a triangular matrix. We find necessary and sufficient conditions for the uniform convergence of the sequence of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence. We generalize and improve the familiar results on the interpolation of functions with finitely many singular points by rational fractions and of entire functions by polynomials.

Keywords

analytic function singularity of a function interpolation process rational fraction uniform convergence convergence conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Goncharov V. L., “On interpolation of functions with finitely many singularities by rational functions,” Izv. Akad. Nauk SSSR Ser. Mat., No. 2, 171–189 (1937).Google Scholar
  2. 2.
    Lipchinskiĭ A. G., “Interpolation of analytic functions with finitely many singularities,” Siberian Math. J., 53, No. 5, 821–838 (2012).zbMATHCrossRefGoogle Scholar
  3. 3.
    Polya G. and Szegö G., Problems and Theorems in Analysis. Vol. 1: Series, Integral Calculus, Theory of Functions, Springer-Verlag, Berlin, Heidelberg, and New York (1998).CrossRefGoogle Scholar
  4. 4.
    Gel’fond A. O., Calculus of Finite Differences [in Russian], Nauka, Moscow (1967).Google Scholar
  5. 5.
    Ibragimov I. I. and Keldysh M. V., “On interpolation of entire functions,” Mat. Sb., 202, 283–292 (1947).Google Scholar
  6. 6.
    Ibragimov I. I., Function Interpolation Methods and Some of Their Applications [in Russian], Nauka, Moscow (1971).Google Scholar
  7. 7.
    Goncharova M. K., “On some interpolation problems that are a generalization of the Newton and Stirling series,” Uchen. Zap. Mosk. Univ., 30, 17–48 (1939).Google Scholar
  8. 8.
    Dvorkin V. S., “The Newton interpolation problem for an entire function with special interpolation nodes,” Tr. Stavropol’ Ped. Inst., No. 10, 67–75 (1958).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.P. P. Ershov Ishim State Pedagogical Institute IshimTyumen’ RegionRussia

Personalised recommendations