Skip to main content
SpringerLink
Log in

Inverse Polynomial Images are Always Sets of Minimal Logarithmic Capacity

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

In this paper, we prove that each inverse polynomial image (that is, each inverse image of an interval with respect to a polynomial mapping) is a set of minimal logarithmic capacity in a certain sense. Such sets play an important role in the theory of Padé-Approximation. The proofs are all based on the characterization theorems of Herbert Stahl.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Gonchar, A.A.: Some recent convergence results on diagonal Padé approximants. In: Approximation Theory, vol. V (College Station), pp. 55–70 (1986)

  2. Gonchar, A.A., Rakhmanov, E.A.: Equilibrium distributions and the rate of rational approximation of analytic functions. Math. USSR Sb. 62, 305–348 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Nuttall, J.: Sets of minimal capacity, Padé approximants and the bubble problem. In: Bardos, C., Bessis, D. (eds.) Bifurcation Phenomena in Mathematical Physics and Related Topics, pp. 185–201. Reidel, Dodrecht (1980)

  4. Nuttall, J., Singh, S.R.: Orthogonal polynomials and Padé approximants associated with a system of arcs. J. Approx. Theory 21, 1–42 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Peherstorfer, F.: Minimal polynomials for compact sets of the complex plane. Constr. Approx. 12, 481–488 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Peherstorfer, F., Schiefermayr, K.: Description of inverse polynomial images which consist of two Jordan arcs with the help of Jacobi’s elliptic functions. Comput. Methods Funct. Theory 4, 355–390 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)

  8. Schiefermayr, K.: Geometric properties of inverse polynomial images. In: Proceedings Approximation Theory XIII: San Antonio 2010. Springer Proceedings in Mathematics, vol. 13, pp. 277–287 (2012)

  9. Schiefermayr, K.: Chebotarev’s problem and inverse polynomial images. Acta Math. Hung. 214, 80–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schiefermayr, K.: A density result concerning inverse polynomial images. Proc. Am. Math. Soc. 142, 539–545 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stahl, H.: Sets of extremal capacity and extremal domains.arXiv:1205.3811 (2012)

  12. Stahl, H.: Extremal domains associated with an analytic function, I, II. Complex Var. Theory Appl. 4, 311–324, 325–338 (1985)

  13. Stahl, H.: The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91, 139–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank the referee for pointing out an error in an earlier version of the paper and for several important suggestions.

Author information

Authors and Affiliations

  1. School of Engineering and Environmental Sciences, University of Applied Sciences Upper Austria, Stelzhamerstr. 23, 4600, Wels, Austria

    Klaus Schiefermayr

Authors
  1. Klaus Schiefermayr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Schiefermayr.

Additional information

Communicated by Stephan Ruscheweyh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schiefermayr, K. Inverse Polynomial Images are Always Sets of Minimal Logarithmic Capacity. Comput. Methods Funct. Theory 16, 375–386 (2016). https://doi.org/10.1007/s40315-015-0143-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-015-0143-x

Keywords

Mathematics Subject Classification

Search

Navigation